Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
UNIVERSITA DEGLI STUDI DI PARMA
DOTTORATO DI RICERCA IN FISICA
CICLO XXXI
Dynamical processes and memory effects on temporal networks
CoordinatoreChiarmo Prof Cristiano Viappiani
TutorChiarma Prof Raffaella Burioni
DottorandoMichele Tizzani
Anni 20152018
ABSTRACT 2
Abstract
Understanding the mechanism behind the diusion of a disease has always been a
crucial problem for health and society In particular the evaluation of the threshold
above which there is an emergence of an epidemic state is one of the fundamental
problems for disease control From the mathematical point of view many models
have been formalized over the years In particular the introduction of networks
science has signicantly improved both the analysis and the prediction capability
of these phenomena giving more compelling results Many of the studies in epi-
demics have been carried on static networks but in the last few years the interest
toward time-varying networks has rapidly grown Despite the increased complexity
from the introduction of the time variable some models allow an analytical un-
derstanding of the spreading processes In particular in our work we will consider
the activity-driven model in which the time variable is embedded in the propensity
of each individual to interact at a certain time Social ties are also driven by the
memory people have of each other preferring old acquaintances interactions to new
ones In this work we explore the eects that the memory mechanism in a time-
varying network has on a spreading process focusing on the epidemic dynamics
We focus our attention on two standard epidemic models the susceptible-infected-
susceptible (SIS) and the susceptible-infected-recovered (SIR) describing respec-
tively diseases that dont or do confer immunity after the infection We formulate
an activity-based mean-eld approach obtaining analytically the epidemic thresh-
old as a function of the parameters describing the distribution of activities and the
strength of the memory eects In particular we consider the asymptotic regime
in which the infection starts only when the people have had a suciently large
number of connections in their social circle In this limit the dynamical process
can be seen as an activity-driven process evolving on an eective static graph Our
results show that memory amplies the activity uctuations reducing the thresh-
old and enhancing the epidemic spreading in both the SIS and SIR models To
numerically prove our ndings we simulate the epidemic process on both the time-
evolving and the eective static networks varying the memory parameter and the
ABSTRACT 3
starting time of the infection Comparing the theoretical model with the numeri-
cal simulations we conrm our predictions in the asymptotic limit We also show
that in the preasymptotic regime there are strong aging eects making the epi-
demic threshold deeply aected by the starting time of the outbreak In particular
for short starting times of the infection the correlations induced by the memory
produce strong backtracking eect in both the SIS and SIR processes lowering
or increasing the epidemic threshold respectively We discuss in detail the origin
of the model-dependent preasymptotic corrections setting the bases for potential
epidemics control methods on correlated temporal networks
Contents
Abstract 2
Preface 5
Introduction 6
Chapter 1 Static networks 10
11 Basic denitions of static networks 10
12 Properties of complex networks 16
13 Models of Complex Networks 19
14 Random walk on static networks 24
Chapter 2 Temporal networks 27
21 Representing temporal network 27
22 Statistical properties of TVN 31
23 Models of temporal networks 32
24 Activity-driven network 33
Chapter 3 Epidemic Models 45
31 Introduction 45
32 Traditional models 45
33 Epidemics on static networks 50
34 Epidemics on time evolving networks 58
Chapter 4 Epidemic Spreading and Aging in Temporal Networks with
Memory 64
41 Introduction 64
42 The model 65
43 Analytical results 66
44 Linear Stability Analysis 72
45 Numerical simulations 72
Chapter 5 Conclusions 79
Bibliography 81
4
PREFACE 5
Preface
The work presented in this dissertation as been carried on at the Department of
Mathematics physics and Computer Science of the University of Parma
The rst three chapters introduce the main subjects of this work which that are
static and time-varying networks and epidemic processes
Chapter 4 is the result of the collaboration with Claudio Castellano Stefano Lenti
Enrico Ubaldi Alessandro Vezzani and Raaella Burioni and it is based on the
paper Epidemic Spreading and Aging in Temporal Networks with Memory Ref
[118]
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
ABSTRACT 2
Abstract
Understanding the mechanism behind the diusion of a disease has always been a
crucial problem for health and society In particular the evaluation of the threshold
above which there is an emergence of an epidemic state is one of the fundamental
problems for disease control From the mathematical point of view many models
have been formalized over the years In particular the introduction of networks
science has signicantly improved both the analysis and the prediction capability
of these phenomena giving more compelling results Many of the studies in epi-
demics have been carried on static networks but in the last few years the interest
toward time-varying networks has rapidly grown Despite the increased complexity
from the introduction of the time variable some models allow an analytical un-
derstanding of the spreading processes In particular in our work we will consider
the activity-driven model in which the time variable is embedded in the propensity
of each individual to interact at a certain time Social ties are also driven by the
memory people have of each other preferring old acquaintances interactions to new
ones In this work we explore the eects that the memory mechanism in a time-
varying network has on a spreading process focusing on the epidemic dynamics
We focus our attention on two standard epidemic models the susceptible-infected-
susceptible (SIS) and the susceptible-infected-recovered (SIR) describing respec-
tively diseases that dont or do confer immunity after the infection We formulate
an activity-based mean-eld approach obtaining analytically the epidemic thresh-
old as a function of the parameters describing the distribution of activities and the
strength of the memory eects In particular we consider the asymptotic regime
in which the infection starts only when the people have had a suciently large
number of connections in their social circle In this limit the dynamical process
can be seen as an activity-driven process evolving on an eective static graph Our
results show that memory amplies the activity uctuations reducing the thresh-
old and enhancing the epidemic spreading in both the SIS and SIR models To
numerically prove our ndings we simulate the epidemic process on both the time-
evolving and the eective static networks varying the memory parameter and the
ABSTRACT 3
starting time of the infection Comparing the theoretical model with the numeri-
cal simulations we conrm our predictions in the asymptotic limit We also show
that in the preasymptotic regime there are strong aging eects making the epi-
demic threshold deeply aected by the starting time of the outbreak In particular
for short starting times of the infection the correlations induced by the memory
produce strong backtracking eect in both the SIS and SIR processes lowering
or increasing the epidemic threshold respectively We discuss in detail the origin
of the model-dependent preasymptotic corrections setting the bases for potential
epidemics control methods on correlated temporal networks
Contents
Abstract 2
Preface 5
Introduction 6
Chapter 1 Static networks 10
11 Basic denitions of static networks 10
12 Properties of complex networks 16
13 Models of Complex Networks 19
14 Random walk on static networks 24
Chapter 2 Temporal networks 27
21 Representing temporal network 27
22 Statistical properties of TVN 31
23 Models of temporal networks 32
24 Activity-driven network 33
Chapter 3 Epidemic Models 45
31 Introduction 45
32 Traditional models 45
33 Epidemics on static networks 50
34 Epidemics on time evolving networks 58
Chapter 4 Epidemic Spreading and Aging in Temporal Networks with
Memory 64
41 Introduction 64
42 The model 65
43 Analytical results 66
44 Linear Stability Analysis 72
45 Numerical simulations 72
Chapter 5 Conclusions 79
Bibliography 81
4
PREFACE 5
Preface
The work presented in this dissertation as been carried on at the Department of
Mathematics physics and Computer Science of the University of Parma
The rst three chapters introduce the main subjects of this work which that are
static and time-varying networks and epidemic processes
Chapter 4 is the result of the collaboration with Claudio Castellano Stefano Lenti
Enrico Ubaldi Alessandro Vezzani and Raaella Burioni and it is based on the
paper Epidemic Spreading and Aging in Temporal Networks with Memory Ref
[118]
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
ABSTRACT 3
starting time of the infection Comparing the theoretical model with the numeri-
cal simulations we conrm our predictions in the asymptotic limit We also show
that in the preasymptotic regime there are strong aging eects making the epi-
demic threshold deeply aected by the starting time of the outbreak In particular
for short starting times of the infection the correlations induced by the memory
produce strong backtracking eect in both the SIS and SIR processes lowering
or increasing the epidemic threshold respectively We discuss in detail the origin
of the model-dependent preasymptotic corrections setting the bases for potential
epidemics control methods on correlated temporal networks
Contents
Abstract 2
Preface 5
Introduction 6
Chapter 1 Static networks 10
11 Basic denitions of static networks 10
12 Properties of complex networks 16
13 Models of Complex Networks 19
14 Random walk on static networks 24
Chapter 2 Temporal networks 27
21 Representing temporal network 27
22 Statistical properties of TVN 31
23 Models of temporal networks 32
24 Activity-driven network 33
Chapter 3 Epidemic Models 45
31 Introduction 45
32 Traditional models 45
33 Epidemics on static networks 50
34 Epidemics on time evolving networks 58
Chapter 4 Epidemic Spreading and Aging in Temporal Networks with
Memory 64
41 Introduction 64
42 The model 65
43 Analytical results 66
44 Linear Stability Analysis 72
45 Numerical simulations 72
Chapter 5 Conclusions 79
Bibliography 81
4
PREFACE 5
Preface
The work presented in this dissertation as been carried on at the Department of
Mathematics physics and Computer Science of the University of Parma
The rst three chapters introduce the main subjects of this work which that are
static and time-varying networks and epidemic processes
Chapter 4 is the result of the collaboration with Claudio Castellano Stefano Lenti
Enrico Ubaldi Alessandro Vezzani and Raaella Burioni and it is based on the
paper Epidemic Spreading and Aging in Temporal Networks with Memory Ref
[118]
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
Contents
Abstract 2
Preface 5
Introduction 6
Chapter 1 Static networks 10
11 Basic denitions of static networks 10
12 Properties of complex networks 16
13 Models of Complex Networks 19
14 Random walk on static networks 24
Chapter 2 Temporal networks 27
21 Representing temporal network 27
22 Statistical properties of TVN 31
23 Models of temporal networks 32
24 Activity-driven network 33
Chapter 3 Epidemic Models 45
31 Introduction 45
32 Traditional models 45
33 Epidemics on static networks 50
34 Epidemics on time evolving networks 58
Chapter 4 Epidemic Spreading and Aging in Temporal Networks with
Memory 64
41 Introduction 64
42 The model 65
43 Analytical results 66
44 Linear Stability Analysis 72
45 Numerical simulations 72
Chapter 5 Conclusions 79
Bibliography 81
4
PREFACE 5
Preface
The work presented in this dissertation as been carried on at the Department of
Mathematics physics and Computer Science of the University of Parma
The rst three chapters introduce the main subjects of this work which that are
static and time-varying networks and epidemic processes
Chapter 4 is the result of the collaboration with Claudio Castellano Stefano Lenti
Enrico Ubaldi Alessandro Vezzani and Raaella Burioni and it is based on the
paper Epidemic Spreading and Aging in Temporal Networks with Memory Ref
[118]
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
PREFACE 5
Preface
The work presented in this dissertation as been carried on at the Department of
Mathematics physics and Computer Science of the University of Parma
The rst three chapters introduce the main subjects of this work which that are
static and time-varying networks and epidemic processes
Chapter 4 is the result of the collaboration with Claudio Castellano Stefano Lenti
Enrico Ubaldi Alessandro Vezzani and Raaella Burioni and it is based on the
paper Epidemic Spreading and Aging in Temporal Networks with Memory Ref
[118]
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
Introduction
We live in a complex and interconnected world Complex systems from micro-
scopic to macroscopic scale are formed of many interacting elements From atoms
to chemical compounds from cells to organisms from people to society the inter-
actions among the elements of each system denes a hierarchy of complexity that
spans through dierent elds of science From these examples we can see that the
common characteristics of a complex system are that they are composed of a large
number of interacting agents exhibiting emergence ie a self-organized collective
behavior not discernible from the single action of the agents
In the last few years the need for a new language to describe complexity has lad
to the science of complex networks There are many examples of complex networks
around us We could dene two main classes of real networks infrastructures
an natural systems [12] In the rst category we found virtual structures like
the World Wide Web or physical structures like power greed and transportation
networks On the other hand we can refer to natural networks as to the structures
forming or form by living entities like biological and social systems As we can
see networks science touches dierent subjects and for this reason most of the
denitions describing the network are borrowed for dierent scientic elds
The network paradigm can be very useful to study dynamical processes such as
information diusion or epidemic spreading which can be seen as additional ingre-
dients evolving on top of the network structure This approach allows studying the
interplay that exists between the dynamical process and the structure from both
sides
The rst approach to study dynamical system on complex networks is to consider
a static approximation of the graph where the time-scales of the evolution of the
network are either too slow or too fast respect to the dynamics of the process on
top of it
On the other hand in most social and information systems time scales of networks
dynamics are often comparable to the time scales of the dynamical processes taking
place on top of them The diusion of online information and the spreading of
transmitted diseases in a population are typical examples of such processes In
these cases the static representation of the network is not able to grasp all the
features of the rapidly changing topology [10 38 2 9 124] Modern technologies
are able to measure and monitor the evolution of interactions with a high time
6
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
INTRODUCTION 7
Figure 001 Metabolic Network Orange nodes carbohydrate metab-
olism Violet nodes photosynthesis Metabolic metro Red nodes cellular
respiration Pink nodes cell signaling Blue nodes amino acid metabo-
lism Grey nodes vitamin and cofactor metabolism Brown nodes nu-
cleotide and protein metabolism Green nodes lipid metabolism Source
httpsenwikipediaorgwikiMetabolic_network
Figure 002 World Wide Web map visualization of routing paths
through a portion of the Internet Source httpsenwikipediaorgwiki
Internet
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
INTRODUCTION 8
resolution [29] calling for new theories to understand the eect of time-varying
topologies on dynamical processes
Especially in social systems the time evolution of the contacts is the result of the hu-
man activity a quantity that can be easily measured from the available large scale
and time-resolved datasets [101] This analysis shows that human activities are
typically highly heterogeneously distributed and this has strong eects on network
evolution To explicitly include the eect of activity distributions on the network
dynamics it has been introduced the activity-driven networks [94] In this frame-
work each agent is endowed with a degree of freedom that encodes the propensity
of the individual to engage in a social event establishing a link with another agent
in the system
When links are randomly established among agents activity-driven models have
been studied in detail [95 94 111 99] uncovering the eects of heterogeneous
activity distributions on network topology and on dynamical processes such as
random walks and epidemic processes
However in general agents do not connect randomly to their peers [45 72 102]
During their activity individuals remember their social circles and they are more
inclined to interact with their history of contacts establishing strong and weak ties
with their peers [36 116] Recently this problem has been tackled by applying a
data-driven approach and measuring the tie allocation mechanism in real systems
introducing a memory process to activity-driven models[55 60] As reasonably
expected social interactions are not randomly established but they are rather con-
centrated towards already contacted nodes with a reinforcement process encoded
in a single measurable memory parameter The memory process tunes the network
evolution that can be predicted at large times [121 23 59] and it is also expected
to inuence dynamical processes Indeed it has been shown that it changes the
spreading rate in a diusion process slowing it down in some cases and speeding
it up in others [100 103 64 62 53 54 98] Similarly in epidemic spreading on
activity-driven networks it can be shown that memory can lower or increase the
epidemic threshold in SIS or SIR model respectively [117] This happens when
the epidemic process and the network evolution start at the same time However
in presence of a memory process as observed also in other elds[47] the network
evolution could introduce aging in the process [73] and this could further inuence
the spreading dynamics In our work we analyze these phenomena giving a full
understanding of their eects on the epidemic dynamics
We formulate the activity-based mean-eld model and analytically derive the epi-
demic threshold as a function of the memory parameter and the activity distribution
for both the SIS and the SIR models In particular we consider the asymptotic limit
in which the epidemic process starts when the individuals have already reached a
certain average number of contacts in their social circle In this regime we can con-
sider the epidemic evolution as a dynamical process evolving on an eective static
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
INTRODUCTION 9
network The analytic results show that the memory amplies the activity uctu-
ations lowering the the epidemic threshold respect to the memoryless case for both
the SIS and the SIR processes We compare the analytical model with numerical
simulation on both the time-evolving and the eective static networks conrming
our predictions
The aging eects are recovered in the preasymptotic regime when for short start-
ing time of the infection the memory induces correlations among the infection
probabilities of the nodes already contacted Because of these correlations both
the SIS and SIR present backtracking eects which lower or increase the epidemic
threshold respect to the mean-eld result In this work we will discuss the reasons
of this deviation opening new horizons for controlling and understanding disease
and information spreading in networks with high correlations
This work is organized as follows In Chapter 1 we will introduce the basic con-
cepts and models of static networks In Chapter 2 we will introduce time-varying
networks and in particular the activity-driven framework In Chapter 3 we will
explore the main models of epidemic spreading on both static and time-varying
networks Chapter 4 is the results of original research on the epidemic spreading
in time-varying networks with memory
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml
CHAPTER 1
Static networks
In this chapter we will introduce the basic concepts and models of static networks
[83 129 4] that will be useful to understand some of the results on epidemic
models described in the third chapter In this framework the structure of the
system doesnt evolve in time and we will see later in this work how this property
aects the dynamics of a process evolving on the network
In the rst section we will introduce the basic denition of the graph theory while in
the second section we will consider the statistical properties of complex networks In
section 13 we will describe some of the principal models of complex static networks
and nally in the last section of we will introduce the random walk formulation on
static networks
11 Basic denitions of static networks
The natural theoretical framework to study complex networks is graph theory [18
21 4] A graph or a network G = (VL) is a structure consisting in a set of
vertices or nodes V and a collection of links or edges L such that V is non null
and L is formed by pairs of elements of V A subset Gprime(V prime Lprime) of a graph G is
called subgraph if V prime sub V and Lprime sub L
The number of elements N in V represents the order of the network while the
number of element in L ie the total number of links is denoted by K and
represents the size of the network so that it is possible to dene a graph also by
its order and size G(NK)
In a network G(VL) two nodes i and j are said to be adjacent or neighboring if
there is a link between them this can be expressed in the matrix representation by
the adjacency matrix A which elements Aij are dened as follows
Aij =
1 (i j) isin L
0 (i j) isin L
We can introduce the main denitions of a network according to the properties of
A and of the sets V and L
bull If we associate a real number wij to an edge between two nodes i and
j the graph is weighted (Figure 111(d)) An example is given by the
dierent strengths of social ties [13] in biological systems like food webs
10
11 BASIC DEFINITIONS OF STATIC NETWORKS 11
where weights represents dierent carbon ows between spices [66] or in
transportation networks where they represent the trac ow [87]
bull If a node i can be connected to itself Aii 6= 0 we have a loop or self-
edge (gure 111(f)) In this case an example is given by the network
of transcription interactions in the EColi bacteria where the self-edge is
the transcription factors that regulate the transcription of their own genes
[130]
bull If two nodes i and j are connected with more than one link Aij gt 1 we
have multiedges and the graph is a multigraph (gure 111(c))
bull If the sets of nodes and links are unordered the the graph is undirected
(gure 111(a)) in this case two adjacent nodes i and j are mutually con-
nected Aundirected Eq111 Most of the graphs we will consider from now
on belong to this category The simplest example to picture is friendship
which is a mutual interaction
bull If the set of nodes and links is ordered the graph is directed (gure 111(b))
which mean that the edge between two nodes has a dened direction and
generally nodes are not mutually connected Adirected Eq112 An ex-
ample of this type of connection is given by the citation networks [109]
where because of the temporal order of the publications a cited article
cant cite back the source of the citation
(111) Aundirected =
0 1 0
1 0 1
0 1 0
(112) Adirected =
0 1 0
0 0 1
1 0 0
bull The number of links attached to a node i denes the degree ki of the node
(gure 112)
In an undirected graph the degree can be expressed in term of the adja-
cency matrix as
ki =
Nsumj=1
Aij
in this case every edge has two end so that the total number of edges is
K =1
2
Nsumi=1
ki
11 BASIC DEFINITIONS OF STATIC NETWORKS 12
(a) Simple undirected graph (c) Muledge graph(b) Directed graph
w
(d) Weighted graph (e) Complete graph (f) Self-loop graph
Figure 111 (a) Simple undirected graph (b) directed graph(c) multiedge graph (d) weighted graph (e) complete graph (f)self-loop graph
For a directed graph we need to distinguish between the outgoing and
the incoming edges of the node i dening respectively the in-degree (g-
ure 112(b)) kini and the out-degree (gure 112(c)) kouti
kin
i =
Nsumj=1
Aji
kouti =
Nsumj=1
Aij
the total in-going number of edges is equal to the total out-going edges
K =
Nsumi=1
kini =
Nsumj=1
kouti
hence the mean in-degree is equal to the mean out-degree
For weighted networks we can dene a weighted degree kwi for a node i
given by
kwi =
Nsumj=1
Awij
We can also dene the strength of as the generalization of the degree for
weighted networks as
si =sumj
wij
where the sum is over all the neighbors of i
We will not go in further details with weighted networks and from now
on we will refer only to undirected graphs except when specied
11 BASIC DEFINITIONS OF STATIC NETWORKS 13
(a) Degree K2=5 (b) In-degree K2in=2(c) Out-degree K2out=3
Figure 112 Degree for undirected (a) and directed (b) (c) networks
bull The connectance or density κ of the graph is the ratio between the total
number of links K and the maximum number of possible links Kmax
κ =K
Kmax=
K(N
2
)A graph G is sparse if K N2 ie κ rarr 0 for N rarr infin dense if
K = O(N2) and κ is constant in the limit of N rarr infin or complete if
K = Kmax =(N2
)= N(N minus 1)2 ie the nodes are all connected to-
gether When analyzing or simulating sparse networks it is computational
convenient to dene the adjacency list which given a node i is the set
l = (i s isin L(i)) of all its rst neighbors
bull The k-core of a graph G is the biggest subgraph in which all the nodes
have at least degree k
To understand how to move across a network we need to introduce further deni-
tions that characterize the metric of the system
bull A series of consecutive edges connecting i0 to in through n edges is a path
Pi0in of length n which mathematically speaking is subgraph Gprime(V prime Lprime)
of an ordered collection of n+ 1 vertexes V prime and n edges such that is isin Vand (isminus1 is) isin L for all s When a path passes once through all the
nodes not necessary using every edges it is an Hamiltonian path On
the other hand if a path passes through all the edges but not necessary
through all the nodes it is an Eulerian path [83]
bull A closed path forms a circuit when i0 = in or a cycle if all nodes of the
circuit are distinct circuits from Hmiltonian or Eulerian paths are called
Hamiltonian or Eulerian circle respectively A set of k connected nodes
without a cycle forms a tree of order k and a set of disconnected trees
form a forest A tree of order k with maximum diameter 2 forms a star
bull If there is a path between every couple of nodes the graph is sad to be
connected and the property of being connected is the connectivity
bull A connected subgraph forms a component while a complete subgraph
forms a clique
bull A component that scales as the size of the network N diverging in in the
innite size limit is called giant component [20]
11 BASIC DEFINITIONS OF STATIC NETWORKS 14
bull If it is possible to divide a graph in n classes such that all the vertexes
in the same class are not adjacent the graph is called n-partite graph In
the special case of n = 2 we have a bipartite graph [7] An example of
bipartite graph is given by the aliation network in which a two sports
clubs share the same player during two season of a championship [51]
bull A tree is a connected graph without cycles while a forest is a not con-
nected acyclic graph ie composed by multiple trees The natural social
example of a tree is the genealogy graph which is also directed if we
consider the relation of being son to the next node
bull The distance dij between two nodes i and j is the shortest path length
to travel from i to j and is given by
dij = min
sumklisinPij
Akl
Another denition of distance can be introduced substituting the adja-
cency matrix Akl with Aminus1kl 6= 0 which in case of simple graphs with
entries either 1 or 0 make equal sense but for weighted graph
dwij = min
sumklisinPij
Awkl
and
dwij = min
sumklisinPij
[Awkl]minus1
have dierent meaning
bull The diameter of a graph is the maximum distance between two nodes
D = maxij
dij
for example we could ask what is the diameter of the World Wide Web
[5]
bull The average shortest path length or characteristic path length is the aver-
age geodesic distance over all couple of nodes
〈d〉 =1
N(N minus 1)
sumij
dij
bull Some time is more convenient to use the harmonic mean of the distance
introducing the eciency [63] which denes how eciently a network
exchange information
〈e〉 =1
N(N minus 1)
sumij
[dij ]minus1
11 BASIC DEFINITIONS OF STATIC NETWORKS 15
(b) Path of length 4(a) Cycle
Figure 113 Cycle (a) and path (b) for an undirected graph
Biparte Graph
Tree Clique
Figure 114 Bipartite graph clique and tree
To study the importance of a node i in a network we introduce some of the main
centrality measures These indicate for example how inuential is an individual in
a social network or help to identify super-spreaders in epidemic processes
bull The simplest centrality measure is the degree centrality dened by the
degree of the node i
bull The closeness centrality is the average shortest path from the node i to
all the others The more a node is close to the others the more is central
gi =1sum
i6=j dij
bull The betweenness centrality quanties the capability for a node to be a
bridge between the others The more edges componing shortest paths
pass through i the more the node is central
bi =sumh6=j 6=i
σhj(i)
σhj
where σhj is the total number of shortest path from h to j and σhj(i) are
the ones that pass through i
We can measure the tendency for the nodes of a graph to be connected between each
other and characterize the local structure of the neighbors of the node i studying
the clustering of the network
bull The clustering coecient is number of links around a node i
Ci =2
ki(ki minus 1)
sumjk
AijAjkAki
12 PROPERTIES OF COMPLEX NETWORKS 16
1 1 1
Figure 115 Cluster coecient for the node 1
for a vertex with ki gt 1 which is the rate between the number of pairs of
connected neighbors of i and number pairs of neighbors of i It measure
the local group cohesiveness
The denition can be extended to directed networks considering the pos-
sible direction of the edges
Cini =2
kini (kini minus 1)
sumjk
AijAjk(Aki +Aki)
2
Couti =2
kouti (kouti minus 1)
sumjk
AijAjk(Aki +Aki)
2
In some occasion can be also useful to evaluate the average clustering
coecient given by
〈C〉 =1
N
Nsumi=1
Ci
12 Properties of complex networks
121 Degree distribution A fundamental statistical quantity to charac-
terize a network is the degree distribution P (k) For undirected graphs it is dened
as the probability that a random chosen node has degree k while for directed graphs
we have to distinguish between in-degree P (kin) and out-degree P (kout) where the
same meaning applies to kin and kout
The nth moment of the distribution is given
〈kn〉 =sumk
knP (k)
or in the continuous limit
〈kn〉 =
intdkknP (k)
while for a directed graph we have
〈knin〉 =sumk
kninP (kin) = 〈knout〉 =sumk
knoutP (kout)
and
〈knin〉 =
intdkkninP (kin) = 〈knout〉 =
intdkknoutP (kout)
12 PROPERTIES OF COMPLEX NETWORKS 17
Correlated and uncorrelated networks The degree distribution completely de-
nes the statistical properties of an uncorrelated network However in most real
networks connectivity patterns present signicant correlations that aect both the
topological properties of the network and the dynamical processes evolving on it
In correlated networks [104 90] the probability that a node with degree k is simul-
taneously connected to n other nodes of degree kprime kprime(n) depends on k and it
is represented by P (kprime kprime(n)|k) In fact in general nodes interacts among each
other respect their intrinsic properties dening specic mixing patterns Lets con-
sider the simplest case of a node with degree k connected to a node with degree kprime
the probability P (kprime|k) must satisfy the normalization condition
(121)sumkprime
P (kprime|k) = 1
and the detailed balance condition
(122) kP (kprime|k)P (k) = kprimeP (k|kprime)P (kprime)
which means that the total number of links from vertexes of degree k to vertexes
of degree kprime must be the same pointing from vertexes of degree kprime to vertexes of
degree k for an undirected graph
Introducing the joint degree distribution P (k kprime) representing the probability that
two connected nodes have degree k and kprime respectively and using the conditions
121 and 122 it is possible to obtain the degree distribution as
P (k) =〈k〉k
sumkprime
P (k kprime)
In this case the network is completely characterized by the degree distribution P (k)
and the rst conditional probability
P (kprime|k) =〈k〉P (k kprime)
kP (k)
and in particular for uncorrelated networks
P (kprime|k) =kprimeP (kprime)
〈k〉
The evaluation of P (kprime|k) for a network of nite sizeN is not easy and sometimes to
better understand the mixing topological properties of the network it is convenient
to introduce the average nearest neighbors degree of a node i as
knni =1
ki
Nsumj=1
Aijkj
and the average degree of the nearest neighbors with degree k as
knn(k) =sumkprime
P (kprime|k)kprime
12 PROPERTIES OF COMPLEX NETWORKS 18
(a) Assortave (b) Disassortave (c) No structure
Figure 121 Average degree of neighbors in the assortative (a)disassortative (b) and non-structured (c) cases
In absence of degree correlations knn(k) is a constant
knn(k) =
langk2rang
〈k〉and it is independent from k
If the system presents degree correlations then when knn(k) is an increasing func-
tion of k the graph is called assortative (gure 121(a)) while if it is a decreasing
function of k it is disassortative (gure 121(b))[79] In assortative networks the
nodes tend to connect to their connectivity peers while in disassortative networks
nodes with low degree are more likely connected with highly connected ones
To study the clustering of the network including the correlations among vertexes
can be useful to dene the clustering spectrum ie the average clustering coecient
restricted to the nodes of degree class k [127]
C(k) =1
P (k)N
sumi|ki=k
Ci
Homogeneous and heterogeneous networks Another distinction between net-
works according to their degree distribution is between homogeneous or heteroge-
neous In the rst case the functional form of P (k) is light tailed like a Gaussian
or a Poissonian while in the second case P (k) is heavy tailed The peculiarity of
heavy-tailed distributions is that the average degree does not represent any special
value for the distribution because even if a random choice will typically pick a
node with low degree the probability to extract a node with large degree is still
signicant This property in the absence of an intrinsic scale for the degrees uc-
tuations denes the scale-free networks [25 3] As a matter of fact if the degrees
distribution is power-law
P (k) = Bkminusγ
and considering 2 lt γ le 3 the average degree is well dened and bounded as
〈k〉 =
int infinkmin
kP (k)dk
13 MODELS OF COMPLEX NETWORKS 19
while the second moment langk2rang
=
int infinkmin
k2P (k)dk
diverges and the uctuation of the degree that in this case depends on the size of
the system is unbounded The heterogeneity properties translates in a high level
of degree uctuations and in the absence a characteristic scale for the degree A
parameter to identify the scale-free behavior can be dened as
κ =
langk2rang
〈k〉
so that if κ 〈k〉 the network is considered scale-free
Scale-free networks are particularly suited to describe several real-world networks
[31] For example the presence of hubs nodes with degree highly exceeding the
average in many real systems is a clear manifestation of this property [22 1]
As we will see in this dissertation the dierence between heterogeneous and homo-
geneous networks play a fundamental role in the studying of dynamical processes
evolving on the network
122 Small world Travers and Millgram in the 1960s [119] in their exper-
iment studied how many people are needed to handout a letter passed from person
to person to reach a given target From the results we have the famous six degrees
of separation theory asserting that most people in the world are connected by short
paths of length six
The average shortest path length introduced before is an indicator of how far from
each others nodes are in a network In particular when 〈d〉 sim logN or shorter the
network has the small-world propriety
Despite being a characteristic of random graphs as we will see later this property
has been observed in many real networks and it is associated with a certain level
of clustering as we will see in the Watts and Strograts model [132]
13 Models of Complex Networks
In this section we will present some of the generative models of static networks All
the possibility to create a static graph described here relies on the denition of the
particular choice for probability to form a connection between two nodes
131 Random networks
1311 Erdoumls and Reacutenyi (ER) graph The most simple model for complex net-
works is the Erdoumls and Reacutenyi (ER) random graph model [37] We can dene two
generative processes for this model forming two kind of ensemble GNK and GNpthat are directly related to the canonical and gran canonical ensemble in statistical
mechanics [18]
13 MODELS OF COMPLEX NETWORKS 20
In the rst and original formulation the pairs of nodes forming a numberK of edges
are randomly chosen among the total number of nodes N In the second scenario
the rule to build the network is that each of the N(N minus 1)2 total number of edges
is created with probability p In particular the number of possible representation
of GNp is 2N(Nminus1)1 considering that each edge either exists or not This denes
an ensemble GNp of all the graphs for which the probability of having k edges is
pk(1 minus p)Kminusk consequently we can easily see that the probability that a random
chosen node has degree k is given by the binomial distribution
P (k) =
(N minus 1
L
)pk(1minus p)Nminus1minusk
where L is the total number of links For this model of random graphs many ana-
lytical results can be obtained in the thermodynamic limit N rarrinfin which can also
be extended to the rst model In particular the average degree is constant〈k〉 =
(N minus 1)p and the degree distribution becomes a Poissionian
P (k) =〈k〉k
keminus〈k〉
One of the most important properties of an ER graph is the presence of a phase
transition from low-density with few links and mostly small components to high-
density states in which a large fraction of vertexes are included in a single giant
component The threshold is determined by the critical probability pc = 1N
corresponding to the critical average degree 〈k〉c = 1 For p lt pc and large N the
graph has no component of size grater then O(lnN) with more than one cycle For
p = pc the largest component has size of orderN23 while for p gt pc a component of
size N exists Considering as the order parameter the size of the largest component
this transition is of the second order and it is in the same universality class of the
mean eld percolation phase transition
In the ER graphs the average clustering coecient is 〈C〉 = p = 〈k〉 N which
means that it decreases with the graph size for xed 〈k〉 Moreover the average
shortest path length is given by
〈d〉 sim lnN
ln 〈k〉giving the ER graphs the small-world property
1312 Conguration model The natural extension of the ER model is to con-
sider instead of the Poisson distribution an arbitrary distribution P (k) for the de-
gree To this end we introduce the conguration model dened as the ensemble
of graphs formed by congurations generated with the following recipe [80] First
we assign to each node i a degree ki representing the number of half-links called
stubs choosen from a xed degree sequence D = ki for i = 1 N such that
Nsumi=1
ki = 2L
13 MODELS OF COMPLEX NETWORKS 21
where L is the number of links and the fraction of nodes with degree k tends to
P (k) for large N Second we join together pairs of stubs randomly accordingly to
their degree
This process generate with equal probability every possible graphs compatible with
the xed degree sequence of which each conguration can be obtained inprodi ki
ways considering that the number of permutation of the stubs of a node i is ki
From the generating process two main limitation of the conguration model stand
out the sum of all the ki must add up to an even number and it is not possible to
avoid the formation of self-edges and multiedges
In this framework the probability pij that two nodes i and j are connected given
by
pij =kikj
2Lminus 1
ie the probability for the node i to connect to a node j with degree kj kj(2Lminus1)
times the number of stubs ki attached to i Notice that in the limit of large m the
probability becomes
pij =kikj2L
Hence in this model a node has lower degree than its typical neighbor describing
the criterion of I have less friends then my friends
The conguration model allows analytical calculation for dierent quantity for in-
stance the clustering coecient is given by
C =
[langk2rangminus 〈k〉
]2N 〈k〉3
which as expected in a random graphs is zero in the limit of N rarr infin On the
other hand considering a highly skewed degree distribution the factorlangk2rang 〈k〉2
can be very large and C is not completely negligible for large nite graphs
Also the presence of a giant component can be evaluated using the Molloy-Reed
criterion [75] ie consider a given degree distribution P (k) when
G =sumk
k(k minus 2)P (k) gt 0
and the maximum degree is not too large a giant component is most likely to
appear On the other hand if G lt 0 and the maximum degree is not too large the
size of the largest component is O(k2max lnN)
Using the generating function formalism [82] Newman et al have also characterized
the average shortest path length given by
〈l〉 =ln(Nz1)
ln(z2z1)+ 1
in the limit of N z1 and z2 z1 where zs is the average number of neighbor at
distance s
13 MODELS OF COMPLEX NETWORKS 22
Another property of the conguration model is that the probability of nding more
than one path between any pair of nodes is O(Nminus1) in case of well-behaved distri-
butions If this property is fundamental for the solution of the model on the other
hand it also not not true for most real networks
1313 Scale-free random graphs A very signicant subclass of random graphs
closer to real networks consists in the scale-free random graphs There are many
generative processes that lead to power-law degree distributed random graphs we
will introduce some of them here and talk more extensively later
The Newmans approach is to use the generating functions method introducing a
degree distribution P (k) sim kminusγ and nding the expression C sim N (3γminus7)(γminus1) for
the clustering coecient In this case for γ lt 73 we can say that there can be
more than one edge between two nodes sharing a common neighbor and hence C
increases with the size of the system NWhile for γ gt 73 C tends to zero for large
graphs
It also possible to generate the graph assigning a uniform probability to all random
graphs with a number of nodes k given by N(k) = eαkminusγ In this case a giant
component exists when γ lt γc sim 347875 [18]
Another possible approach is the tness model introduced by Caldarelli et al [24]
In this case we start with N isolated nodes and consider each node i to have a
tness ηi which is a real number drawn from the tness distribution ρ(η) then for
every couple of nodes i and j a link is extracted with probability pij = f(ηi ηj)
where f is a symmetric function [108] and if it is constant we obtain the ER model
This model generate a power-law P (k) for many tness distribution indeed if we
consider a node with tness η its average degree is
k(η) = N
int infin0
f(η ϕ)P (ϕ)dϕ = NF (η)
and the degree distribution is given by
P (k) =
intdηP (η)δ [k minus k(η)] = P
([Fminus1
(k
N
)partkF
minus1
(k
N
)])considering F (η) to be monotonic in η
Lets give an example considering P (η) = eminusη and
f(η ϕ) = θ [η + ϕminus κ(N)]
where κ is a predetermined threshold and θ is the Heaviside function This process
generates a scale free network with degree distribution P (k) sim kminus2 from a picked
tness distribution In this case both the assortativity knn(k) and the clustering
spectrum C(k) are power-laws
1314 Watts and Strogatz Many real systems have both the small-world prop-
erty and the high clustering coecient one of the simplest method to have them
both was formulated by Watts and Strogatz (WS) [132] The basic feature is a
13 MODELS OF COMPLEX NETWORKS 23
rewiring procedure of the edges with a probability p The generating process con-
siders a ring of N nodes each symmetrically connected to its 2m nearest neighbors
considering that the total number of links is L = mN each edge is then rewired
with probability p or preserved with probability 1minusp moving its endpoint to a new
node randomly chosen from the graph excluding multiedges or loops Notice that
if p = 0 we have a regular lattice with clustering coecient C = (3k minus 3)(4k minus 2)
while for p = 1 we reproduce a random graph with kmin = m with distance of
order logN log k and very low clustering coecient C sim 2kN Hence we can
see p as tuning parameter for the randomness of the graph keeping the number of
edges constant and for 0 lt p lt 1 we generate graphs that have the two wanted
characteristics The clustering coecient in this case is
C =3(k minus 1)
2(2k minus 1)(1minus p)3
Also the degree distribution is exactly computed and has the form
P (k) =
min(kminusmn)sumn=0
(m
n
)(1minus p)npmminusn (pm)kminusmminusn
(k minusmminus n)eminuspm for k ge m
which in the limit of prarr 1 gives us the Poisson distribution as expected
Regarding the average path length we can estimate a number of partial exact results
and some scaling results In the limit of prarr 0 the typical path length is l = N4k
while in the large p limit l sim logN which is the small-world property Bartheacuteleacutemy
and Amaral [14] formulate the following scaling relation for l
l = ξg(Nξ)
where ξ is a correlation length that depends on p and g(x) is an unknown scaling
function that depends only on the system dimension and on the geometry of the
graph and in the two limit situation takes the values
g(x) =
x x 1
log x x 1
they also showed that for small p ξ sim pminusτ where τ is a constant
1315 Preferential attachment The preferential attachment model formu-
lated by Barabasi and Albert (BA) [4] was inspired by the formation of the World
Wide Web and it is characterized by two main ingredients
bull growth which means that starting with m0 isolated nodes at each time
step ∆t = 1 2 N minusm0 a new node j with m le m0 number of links is
added to the network and it will connect to a preexisting node i
bull preferential attachment the probability Π for a new node to connect to
a preexisting node i depends on the degree ki
Π(ki) =kisumj kj
14 RANDOM WALK ON STATIC NETWORKS 24
After t time steps the size of the network will be N = m0 + t and the number of
edges will be mt
The idea is that as for the WWW nodes with high degree form new link with higher
rate than the ones with low-degree In the long time limit this model produces a
power-law degree distribution P (k) sim kminusγ with γ = 3 This result can be easily
obtained considering that each time a new link attaches to a preexisting node i the
degree increases as follows
parttki = mΠ(ki)
where we have considered ki as a continuous variable The last expression can be
written explicating the probability Π(ki) and considering the k limit becoming the
following
parttki(t) =mki(t)
2mt+m0 〈k〉0
where 〈k〉0 is the average degree of the initial m0 nodes Solving for ki with the
initial condition that each node starts at ti with m edges we obtain that
ki(t) m(t
ti
)12
In the large time limit the degree distribution is given by
P (k) = 2m3kminus3
while the clustering coecient is given by
C =m
8N(lnN)2
and the average shortest path is
〈l〉 =logN
log logN
Notice that in the BA model the growth of the network is only in the generative
process and the result is a static network
14 Random walk on static networks
We want to study the evolution of the number of elements passing through a node
using the random walk formalism [68 86]
In the simplest Markovian random walk formalism we study a diusive process on
an undirected unbiased network In a network with N nodes and adjacency matrix
Aij a walker i at times t hops to one of its ki =sumj Aij neighbors chosen with
equal probability at time t+ 1 hence we dene the transition probability πij to go
from i to j as
πij =Aijki
=AijsumNj=1Aij
We want to study the evolution equation of the occupation probability distribution
pi(t) for a node i to be visited at a certain time t which in the discrete time model
14 RANDOM WALK ON STATIC NETWORKS 25
is given by
(141) pj(t+ 1) =
Nsumi=1
πjipi(t)
When Eq 141 holds also for t = 0 and considering Π = πij as the transitionmatrix we can write in the vector form as
p(t+ 1) = Πp(t)
The stationary distribution plowast is given by the xed point solution of the equation
such that plowast = Πplowast For connected graphs containing at least one odd cycle the
Perron-Frobenius theorem guarantees the existence of plowastsuch that
limtrarrinfin
Πtp(0) = plowast
In this case all the occupation probability distributions converge to the stationary
distribution For undirected networks the stationary distribution is given by
plowasti =kisumj kj
while if the network is also unweighted plowasti = kiK where K is the total number
of links In this case the stationary distribution must also follow the detail balance
condition
plowasti πij = plowastjπji
which means that the ow of probability in each direction of the edge must be equal
at the equilibrium
We now want to focus on a method particularly useful in case of heterogeneous net-
works assuming that all the nodes with the same degree are statistically equivalent
As we will discuss in chapter 3 in this approximation nodes are characterized only
by their degree Lets dene the occupation number Wi representing the number
of walkers for the node i and the total number of walkers W =sumiWi The degree
block variable
Wk =1
NP (k)
sumiisink
Wi
where P (k) is the degree distribution and the sum is over all the nodes with degree
equal to k Introducing the transition rate r for a node with degree k to go to a
node of degree kprime we can write the mean-eld equation for the variation in time of
the walkers Wk(t) in each degree class given by
parttWk(t) = minusrWk(t) + ksumkprime
P (kprime|k)r
kprimeWkprime(t)
where the rst term account for the walkers moving out the node with rate r
and the second is the diusive term describing the walkers that moving from the
neighbors visit the node with degree k
14 RANDOM WALK ON STATIC NETWORKS 26
Because we are considering uncorrelated networks the conditional probability P (kprime|k) =
kprimeP (kprime) 〈k〉 and studying the stationary condition parttWk(t) = 0 we obtain the so-
lution
Wk =k
〈k〉W
N
The probability to nd a diusive walker in a node with degree k is consequently
given by
pk =Wk
W=
k
N 〈k〉hence it is more probable for a walker to diuse into a highly connected node
We will use this method to describe epidemic processes on the network in Chapter
3
CHAPTER 2
Temporal networks
In many situations the introduction of the topology is enough to give an insight
on the mechanisms characterizing the system However in most real situations
connections between individuals evolve in time When studying a dynamical process
evolving on a graph we can consider two opposite time-scale limits in which the
static network framework is still a good approximation The rst one is when the
network evolution is much slower than the dynamical process evolving on it The
second one is the opposite limit in which the evolution of the network is much
faster than the time-scales of the dynamical process evolution In this last case
the dynamic unfolds on the annealed static network [114 41] which is described
by a mean-eld version of the adjacency matrix giving a good approximation the
process
However in most real systems such as social systems both the structure of the
graph and the dynamical processes on it evolve on comparable time scales and the
previous two approximations dont apply In this case it is necessary to introduce
the time variable in the network denition introducing the time-varying networks
(TVN) [85 50 49 67]
Most of the properties valid for static networks cant be easily translated to the
TVN This chapter is organized as follows in section 21 we will introduce the
basic denitions of time-evolving networks while in section 22 we will consider
some of their statistical properties In section 23 we will give some examples of
TVN models and in particular in section 24 we will focus on the activity-driven
model which is at the center of this work
21 Representing temporal network
We can describe static networks with either the adjacency matrix or the adjacency
list representation Also for temporal networks there are many possible repre-
sentations of the system We will focus mainly on the event-based and snapshot
representations
Lets consider a TVN formed by a set of nodes N during an observation interval
[0 T ] In the event-based we consider the TVN as an ordered set C of time-
stamped links called events or contacts recorded in the observation interval The
27
21 REPRESENTING TEMPORAL NETWORK 28
time
51
31
54
32
δt
21
Time integrated graph
Figure 211 Event-based representation
general denition of a contact between two nodes is given by the quadruplet
cij(t δt) = (i j t δt)
where 0 le t le T is the time of the interaction and δt is its duration and if
cij(t δt) = cji(t δt) then the contact is symmetric The duration of an event can
be very long and sometimes as we will see later can be broadly distributed but in
many situations is much smaller than the inter-event time and δt can be neglected
obtaining cij(t) = (i j t) This approximation yields to a useful simplication for
both numerical and analytical analysis of TVN
The other representation consists in describing the TVN as a discrete time series of
the network In this case we consider a snapshot Gt representing the instant cong-
uration of the graph at time t The system is then dened by the ordered sequence
of snapshots G[0T ] = G(t0) G(t1) G(T ) called time aggregated graph where
T is the total number of time intervals considered Each snapshot of the system at
time t isin [0 T ] can be fully described by the adjacency index or adjacency matrix
A(t) at time t whose elements are
aij(t) =
1 i and j are connected at time t
0 otherwise
The sequence of adjacency indexes A = A(t0) A(t1) A(T ) forms the time
aggregated adjacency matrix at time T The snapshot model is a discrete time
representation useful to develop a matrix-based model of temporal networks and
allows incorporating the time variable in the mathematical formulation through
the denition of an adjacency tensor With this representation we can study the
macroscopic properties of the TVN seen as a time-evolving structure
Event-based representation at discrete time can turn in the snapshot representation
without loss of information on the other hand the transformation of continuous
time event in snapshots leads to a loss of temporal information due to the dis-
cretization process
211 Measures Walking in a static network from a node i to a node j
depends only on the existence of a set of links between the two nodes On the other
hand in TVN a walker must wait for the times of the intermediary nodes to be
connected to move around the network Moreover the arrow of time establishes
21 REPRESENTING TEMPORAL NETWORK 29
time
G(0) G(1) G(2)
Time integrated graph
Figure 212 Snapshot representation
a causality constraint not allowing the use of past events These properties make
the denition of temporal walk very dierent from the static network framework
In the contact-based representation a temporal walk from a node ni to a node nj is
a contact sequence [(ni n1 t1) (njminus1 nj tj)] ordered in time ti lt tn1 lt lt tj
Similarly in the snapshot representation the contact (nl nl+1 tl+1) is interpreted
as the link of the snapshot G(tl) such that anlnl+1(tl+1) 6= 0
If a walker visits each node between ni and nj we have a temporal path or time-
respecting path from ni to nj Notice that in the aggregated network temporal
walks and paths are always overestimated because of the presence of all the links
created during a time interval
A node nj is reachable from a node ni if there is a temporal walk between them The
set of nodes that can be reached starting from a node ni is called set of inuence of
ni We can also dene the reachability ratio as the average fraction of nodes in the
set of inuence of all nodes and the set of vertexes that reach ni through temporal
paths in a certain time window as the source set of ni The ordered nature of
time-respecting paths set a fundamental dierence with static networks hence in
TVN the reachability is not a transitive property
212 Metric Most TVN contacts have neither the symmetry nor the tran-
sitivity properties hence we can not dene a proper metric nonetheless we can
introduce the concept of distance either from the topological or from the temporal
perspective obtaining three main denitions valid for both the event-based and the
snapshot representations Lets consider a temporal path [(ni n1 t1) (njminus1 nj tj)]
from ni to nj
The topological shortest distance is given by the minimum number of hops h nec-
essary to go from ni to nj along the temporal path
dshort(ni nj t) = minh t1 ge t
The temporal shortest path or the foremost distance is dened as the minimum
amount of time to travel between two nodes
dfore(ni nj t) = mintj minus t t1 ge t
21 REPRESENTING TEMPORAL NETWORK 30
Another distance in the time domain favors the temporal paths with shortest
duration independently from the starting time
dtravel(ni nj t) = mintj minus t1 t1 ge t
For each distance ddef (ni nj t) where def stands for the three denitions we can
introduce the average distance over all the pairs of nodes given by
Ldef =1
N(N minus 1)
sumi 6=j
ddef (ni nj t)
and the diameter of the TVN
Ddef = maxninj
ddef (ni nj t)
Notice that if a point is not reachable then ddef (ni nj t) = infin and to avoid the
divergence of L we can also dene the temporal global eciency as
E =1
N(N minus 1)
sumi 6=j
1
ddef (ni nj t)
We can also measure the recency of an information exchange between two nodes
or the information latency of ni respect to nj at time t
The temporal view φ(ni nj t) that a node ni has about a node nj at time t is the
latest time tprime lt t such that a temporal path starting from nj at tprime reaches ni before
t The information latency is given by the dierence tminusφ(ni nj t) and represents
the time passed since the most updated information about nj was obtained by
ni Considering the information passed to ni from all the network we can dene
the vector clock φ(ni t) = [φ(ni nj t)]j=1N as the list of the most up to date
information that ni has about all the network [61]
213 Components The denition of temporal paths aects also the con-
cepts of connectivity and components in TVN For static networks components
are dened as the sets of nodes for which a path always exists moreover nodes
are either connected or not In particular in undirected graphs the connectivity
is a reexive symmetric and transitive property On the other hand in temporal
graphs the symmetry property doesnt hold anymore In analogy with directed
static graphs two nodes ni and nj in a TVN are strongly connected if the tem-
poral paths connecting ni to nj and vice versa are directed while they are weakly
connected if the temporal paths connecting them are undirected [84] By using the
reachability and the denitions of connectivity it is possible to introduce dierent
type of temporal components OUTT (ni) (INT (ni)) is the temporal out-component
(in-component) of the node ni ie the set of nodes that can be reached from i
(from which i can be reached) in the TVN while the strongly connected SCCT (ni)
(weakly connected WCCT (ni)) component of the node ni is the set of nodes from
22 STATISTICAL PROPERTIES OF TVN 31
which i can be reached and which can be reached (undirectelly) from i in the TVN
[84]
214 Centrality Also for centrality measures we can consider the time-
dependent and the time-independent denition
Two examples of centrality measures based on time-dependent distance are
bull The temporal closeness centrality
CC(ni t) =1
N minus 1
sumi 6=j
1
ddef (ni nj t)
measure how quickly an edge reach other edges on average [88]
bull The temporal betweenness centrality takes in to account the fraction of
shortest time-respecting paths that pass through a focal vertex
CBni(t) =1
(N minus 1)(N minus 2)
sumni 6=nj
sumk 6= j
k 6= i
U(ni t nj nk)
σjk
where σjk is the number of shortest path from nj to nk and U(ni t nj nk)
is the number of temporal shortest paths from ni to nj in which node i is
traversed from the path in the snapshot t [33]
22 Statistical properties of TVN
Lets consider the contact-based representation of the network considering null the
duration of the events The event sequence associated either to each vertex or each
link is given by t1 te where now the time ti and the number of evens e are
properties of a single node or link and not of the whole network In this section
we will see some of the statistical features of this set which will help us to dene
some models of temporal networks
221 Burstiness Many events especially in human behavioral networks
occur in a short period of time generating a burst and tend to be separated by
long time intervals An event sequence generated from a Poisson process lacks of
this property In fact if we dene the inter-event of a node i τi = ti+1 minus ti as thetime between two consecutive events of an edge then a Poisson generating process
give the inter-event distribution
ψ(τ) = σeminusστ
whit σ a parameter On the other hand real data analysis shows that most human
processes have power-law distributed inter-events ie
ψ(τ) sim τminusα
23 MODELS OF TEMPORAL NETWORKS 32
The latest case denes one of the ngerprints of the bursty behavior of real social
systems [42 128 11]
To measure the burstiness of a process we can introduce the coecient of variation
dened as the standard deviation of τi divided by its average
CV =
radic1eminus1
sumeminus1i=1 (τi minus 〈τ〉)2
〈τ〉
where 〈τ〉 = (e minus 1)minus1sumeminus1i=1 τi For a Poisson distribution of inter-event times
CV = 1 while for periodic sequence of events CV = 0 The normalized form of
the CV is called burstiness B
B =CV minus 1
CV + 1which varies between minus1 and 1 For a Poisson process B = 0 while for a periodic
sequence B = minus1 Extremely burtsy behaviors have B = 1 (CV rarrinfin)
Another statistical measure is the local variation LV dened as
LV =3
eminus 1
eminus1sumi=1
(τi minus τi+1
τi + τi+1
)2
While the CV is greatly aected by extreme large inter-event time the LV remains
conned in the interval [0 3) [106]
23 Models of temporal networks
In the last few years the number of studies of dynamical processes on temporal
networks has signicantly grown A variety of dynamical processes can be well
described using this framework In this section we will mention some of the most
relevant further looking into the activity-driven framework which will be extensively
used in the description of epidemic processes on TVN
Social group dynamics models In this model links represent social ties as
facing another individual and it is possible to write a master equation representing
the changing of the number of people in a group of a given size [115 42] This
approach describes the observation the more isolated is an individual from a group
the less it is probable that it will interact with the group and the longer it is
interacting with the group the more it is likely to stay in the group In this case
the TVN representation more suited to model the process is the interval graphs
formalism
Randomized reference models To assess the importance of a topological
feature in empirical networks analysis it is generally useful to have a reference
model to compare the data with In this type of models event sequences of the
original network are randomly shued in a fashion that removes correlations in
the time domain Considering the variety of possible temporal correlations and
time scales for dierent systems there are many way of achieving this purpose
24 ACTIVITY-DRIVEN NETWORK 33
for example switching o selected types of correlation can help to understand their
contribution to the empirical network Because of their purpose of annihilate time
correlation these models are also called temporal null models Considering a con-
tact sequence TVN we can dene some of these models depending on the type
of randomization process If we go over all the edges sequentially and randomly
substitute them with another pair following some restrains then we have the ran-
domized edges model (RE) Another option would be to randomly permute the
contact times while keeping the graph structure and the number of contacts in-
tact in this case we have the randomly permuted times model (RP) We can also
do both the randomization described before destroying all correlations except for
patterns in the contact rate Instead of keeping the set of times of the original
contact sequence just permuting them as in the RP one could assign a random
time in the observation time window of the original empirical data still conserving
the graph structure and the total number of events for each link It is also possible
to randomize the contacts between the edges (RC)
For more details on the topic it is possible to consult the Ref [40]
24 Activity-driven network
Especially in social systems interactions among individuals rapidly change in time
and the time scale of the evolution of the network is comparable to the time scale
of the dynamical process evolving on it In this case we have seen that the snapshot
representation is well suited to describe the system and the topological properties
can be captured by the time-integrated view of the network The activity-driven
model [94] belongs to this category of TVN and it is the result of empirical studies of
human activity from large data sets The main concept is to embed the dynamical
feature of the system in the node itself instead of studying the links and the
topology of the network
In this model to each vertex vi is associated an activity potential ai which is a ran-
dom variable extracted from a given distribution F (a) representing the tendency
of the node to form a certain arbitrary chosen number m of links at time ∆t Hence
in each snapshot the node vi is active with probability ai∆t and the activity can be
considered as a clock determining the temporal interaction patterns of each node
In the limit of ∆trarr 0 activation of a node follows a Poisson process
The activity-driven model is simply enough to allow analytical investigation of
dierent properties of the structure and the dynamic of the network and it is also
able to reproduce the heterogeneity of many real systems
Generating process and properties The system starts with a network
with N nodes each characterized by the activity ai distributed according to F (a)
The generative process considering that all interactions have a constant duration
is the following
24 ACTIVITY-DRIVEN NETWORK 34
bull At each time step t the snapshot Gt starts with all the nodes disconnected
bull With probability ai∆t each vertex activates and connects m edges to
m randomly selected dierent nodes The nodes not activated can still
receive connections from active nodes
bull At the subsequent time step t + ∆t all the edges in the network Gt are
deleted
At this level the model is random and Markovian hence it has no memory of the
previous time steps and the interaction between the nodes are randomly chosen
The activity distribution F (a) completely determines the topological and evolu-
tionary properties of the network
At rst we can consider the properties of each snapshot The number of active
links per unit time is Lt = mN 〈a〉 so that the average degree per unit time is
〈k〉 = 2m 〈a〉 where the two is the consequence of the undirected nature of each
link and 〈a〉 =intaF (a)da is the average activity Snapshots are generally sparse
networks formed by stars with degree k ge m
Time integrated network We are also interested in the properties of the
time integrated network G =⋃Tt=0Gt dened by the union of all the snapshots gen-
erated in T +1 time steps This network is generally dense and if T is long enough
we obtain a complete graph If we consider the integrated network normalized by
the number of snapshot ie the total time then its adjacency matrix is given by
Aij =m(ai + aj)
N
neglecting the probability for an active node to contact another active node From
this relation we can nd the average degree of the integrated network for the node
i given by
(241) ki =
Nsumj=1
Aij = m(ai + 〈a〉)
From the equation 241 noticing the monotonic relation between the degree and the
activity the following expression for the degree distribution holds ρ(k)dk = F (a)da
Hence taking m = 1 in the limit N T 1 and T 2 k 1
ρ(k) sim 1
TF
(k
Tminus 〈a〉
)
The degree distribution of the time integrated network has the same form of the
activity distribution F (a)
From the analysis of big data sets [44 94] it is possible to show that most human
activities are heterogeneous and characterized by a power-law activity distribution
with exponent ν
F (a) =1minus ν
1minus ε1minusνaminusν
24 ACTIVITY-DRIVEN NETWORK 35
where a isin [ε 1] 0 lt ε 1 is a cuto introduced to avoid divergences of the
distribution in the limit of zero activity In this case the degree distribution is
ρ(k) sim kminusν
On the other hand if F (a) = δaa0 then the asymptotic form of the degree distri-
bution is ρ(k) sim δkk0T while the exact form is a Poisson distribution centered at
2Ta0
Starnini and Pastor-Satorras [112] noticed also that unless F (a) is a delta function
the average degree correlation for integrated network is given in the limit of large
k bykTnn(k)minus 1
T 2 〈a〉+ σ2
a
(k
T
)minus1
where σ2a =
langa2rangminus〈a〉2 is the variance of the activity For delta distributed activity
on the other hand
kTnn(k) 1 + 2Ta0
Hence for non constant activity distribution the time integrated network shows a
disassortative mixing behavior at odds with real social networks which are consid-
ered assortative Notice however that in case of power-law distributed activities
with small variance σa (of order ενminus1 for ν lt 3 and order ε2 for ν gt 3) the network
can be considered approximately uncorrelated
In the limit of large k it is also possible to compute the clustering coecient of the
integrated network which in its asymptotic form is given by
c(k)
T 2 〈a〉
T+
2σ2a
N
(k
T
)minus1
which is small compared to random networks
We can also study the eigenspectrum of the time integrated adjacency matrix [110]
Au = Λu
and assuming that the eigenvector has the propertysumNi=1 ui = 1 we can obtain the
eigenvalues
Λplusmn = m
sumNi=1 aiN
plusmn
radicsumNi=1 a
2i
N
where the N minus 2 other eigenvalues are zero As we will see in the next chapter the
largest eigenvalue Λ+ explains also the analytical result obtained for the epidemic
threshold in the activity driven model
241 Master equation To study the characteristics of the time integrated
network we can also use the master equation formalism considering the evolution
of the probability Pi(k t) that a node with activity ai has degree k at time t [120]
24 ACTIVITY-DRIVEN NETWORK 36
We will extensively use this framework to add dierent levels of complexity to the
activity-driven model
2411 Simple activity-driven model In the simple activity-driven model con-
sidering the approximations where ai 1 so that only one site can be active be-
tween two consecutive times and that 1 ki N the discrete-time equation for
Pi(k t) is
Pi(k t+ 1) =
aiN minus kN
Pi(k minus 1 t) + aik
NPi(k t)minus Pi(k minus 1 t)
sumji
ajsumh
Pj(h t)
N+(242)
Pi(k t)sumji
ajsumh
(N minus 1)Pj(h t)
N+ Pi(k t)
sumjsimi
aj + Pi(k t)
1minussumj
aj
The rst term is the probability that the site i is active and a new edge is added
to the system The second term is the probability that the site i is active but
connects to an already linked site The symbolsumji represent the sum over the
nodes j that are still not connected to i Hence the third and the fourth terms
are the probabilities that one of these nodes j is active and either contact or not i
respectively The fth is the probability that one of the nodes already connected to
i (j sim i ) is active and no new link is added to i The last term is the probability
that all the nodes are inactive at time t
We can neglect the second term for k N and obtain the equation
Pi(k t+ 1)minus Pi(k t) = minus (Pi(k t)minus Pi(k minus 1 t))
ai +1
N
sumji
aj
In this approximation considering that 1
N
sumji aj = 〈a〉 the equation in the con-
tinuous time limit for Pi(k t) is
parttPi(k t) = (ai + 〈a〉)(minuspartkPi(k t) + part2
kPi(k t))
In this case the solution is given by
Pi(k t) = [2π(ai + 〈a〉)t]minus12 exp
(minus (k minus ai + 〈a〉 t)2
2t(ai + 〈a〉)
)In the long time regime this equation reduces to a delta function
Pi(k t) = δ(k minus ai + 〈a〉 t)
and the average degree of the time integrated network for a node of activity a at
time t is given by
k(a t) = (a+ 〈a〉)t
Noticing the monotonic relation between the degree and the activity also the degree
distribution is a power law with the same exponent ρ(k) sim kminusν
24 ACTIVITY-DRIVEN NETWORK 37
Figure 241 On the left the experimental curves of the reinforce-
ment probability p(k) from the PRE and the MPC datasets On the
right from the top the measure of the average degree 〈k(at)〉 = k(at)
for dierent activity classes from the PRB dataset below the degree
distribution for the PRA dataset [120]
2412 Memory process In most real systems concerning human interaction
we have memory processes representing the tendency for an individual to recon-
nect with a previously contacted node In TVN this phenomenon is particularly
relevant In fact because of the time evolution of the contacts the reinforcement
process strongly aects the structure of the network In their empirical study of
the datasets from American Physical Society Twitter Mention Network and Mobile
Phone Networks Ubaldi et al [122] measured the probability p(ki) for a node i with
a history of ki number of contacts to establish a new link nding the expression
(243) pi(ki) =
(1 +
kici
)minusβwhere c is the characteristic limit for an individual to maintain multiple contacts
β is the memory parameter and ki is the degree of the node i in the time integrated
graph (241) In this framework when a node is active it contacts a new randomly
chosen node with probability p(k) or randomly connects to a node in its history of
contacts with probability 1minus p(k)
Introducing the memory process in the equation 242 the equation for the Pi(k t)
is given by
24 ACTIVITY-DRIVEN NETWORK 38
Pi(k t+ 1) =
Pi(k minus 1 t)
aipi(k minus 1) +sumji
ajsumh
pj(h)Pj(h t)
N minus h
+(244)
Pi(k t)
ai [1minus pi(k)] +sumji
ajsumh
(1minus pj(h)Pj(h t)
N minus h
)+
Pi(k t)
1minussumj
aj
The node i can go from k minus 1 to k contacts in two ways described by the second
line of the equation The rst is for i to be active and contact a new node with
probability aipi(k minus 1) and the second is for i to be contacted by an active node
j never contacted before Similarly the third line considers that the node i does
not change degree k either because it calls an already contacted node or because
the non contacted nodes call other nodes in the network The last line describe the
situation in which no nodes in the network activate Using the expression 243 we
can write the equation as
Pi(k t+ 1)minus Pi(k t) =aic
βi
(k minus 1minus ci)βPi(k minus 1 t)minus aic
βi
(k + ci)βPi(k t)
minus (Pi(k t)minus Pi(k minus 1 t))sumji
ajsumh
cβj Pj(h t)
(N minus h) (h+ cj)β
and applying the same approximation of the memory-less case we obtain the equa-
tion for the continuous time limit given by
parttPi(k t) = minusa cβi
kβpartkPi(k t) +
aicβi
2kβpart2kPi(k t) +
βaicβi
kβ+1Pi(k t)
+
(minuspartkPi(k t) +
1
2part2kPi(k t)
)intdajF (aj)aj
intdcjρ(cj |aj)
intdhcβjhβPj(h t)
where ρ(cj |aj) is the probability for a node j of activity aj to have memory constant
cj In this case the solution for the master equation for the probability Pi(k t) for
a node i with activity ai to have a degree k at time t in the long time limit and
for k N is given by
Pi(k t) prop exp
[minusA (k minus C(ai ci)t
1β+1 )2
t1
β+1
]Hence the average degree of the time integrated network for the nodes of activity
a at time t is
k(a c t) prop C(a c)t1
β+1
24 ACTIVITY-DRIVEN NETWORK 39
where C(a c) is a constant depending on the activity which follows the recurrence
relationC(a c)
1 + β=
acβ
Cβ(a c)+
intdaprimeF (aprime)
intdcprimeρ(cprime aprime)
aprimecprimeβ
Cβ(a c)
As we will see in the fourth chapter it is possible to consider c = 1 without loosing
generality In the presence of the memory process the relation between the activity
and the degree is given by k prop a1(β+1) and to nd the degree distribution we can
use the relation
ρ(k) sim kβF(k1+β
)dk
In the special case of a power law activity distribution F (a) sim aminusν we have that
ρ(k) sim kminus[(1+β)νminusβ]
We will extensively use these results in the last chapter where we will use this
model to study two epidemic processes on the activity-driven network with memory
Moreover this model can be extended to multiple β as showed in [120]
2413 Burstiness The activity-driven model considered up to now is Poisso-
nian Now we want to study a non-Poissonian AD process (NoPAD) introducing
the burstiness in the system [74 32] In this framework to each individual i is
associated a time dependent activity ai(t) that the node is active for the rst time
at time t The activation of each node is regulated by an inter-event or waiting
time distribution
Ψi(w) = ai(t) exp
[minusint w
0
ai(wprime)dwprime
]assuming that receiving a call from another individual doesnt aect the time be-
tween two activation events In the limit of constant activity the model coincides
with the simple AD
Because the activities depend on the nodes so must the waiting times and we need
to introduce a node dependent parameter ξi such that Ψi(w) = Ψ(w ξi)
The degree distribution of the time integrated network can be found using the
hidden variable formalism [112] dening
ρ(k) =sum~h
H(~h)g(k|~h)
whereH(~h) is the distribution of the hidden variable ~h and g(k|~k) is the conditional
probability that a node with hidden variable ~h has degree k Introducing the
connection probability Π(~hi~hj) to create a link between the nodes i and j it is
possible to write the equation for the generating function g(k|~h) of the conditional
probability
ln g(z|~h) = Nsum~hprime
H(~hprime) ln[1minus (1minus z)Π(~h~hprime)
]
24 ACTIVITY-DRIVEN NETWORK 40
The hidden variable is vectors ~h = (r ξ) where r is the number of activation that
for a node with time t and heterogeneity ξ is distributed according to χt(r|ξ) Re-stricting the analysis to the time interval [0 t] Moinet et al found the approximated
solution for the degree distribution at time t given by
ρt(k) sumξ
η(ξ)χt(k minus 〈r〉t |ξ)
where η(ξ) is the distribution of the heterogeneity ξ of a node and 〈r〉t is the averagenumber of activation of the node at time t
Empirical ndings suggest to consider the special case in which the distribution for
the waiting times is
Ψ(w ξ) = αξ(ξw + 1)minus(1+α)
where 0 lt α lt 1 is the exponent of the distribution When the heterogeneity
parameter is broadly distributed η(ξ) sim (ξξ0)minusbminus1 (b gt α) and in the limit of
k (ξ0t)α the degree distribution is
ρt(k) = (ξ0t)b(k minus 〈r〉t)
minus1minus(bα)
In this framework the distribution of number of activation events χt(r|ξ) aects
the topological properties of the graph for heavy-tailed waiting times distributions
suggesting aging eects aecting the model In fact as showed in Ref [74 32]
introducing the aged degree distribution ρtat(k) where ta is the aging time the
average degree of the network integrated from time ta depends on ta and is given
by
〈k〉tta sim (ta + t)α minus tαa
Notice that in the limit t ta the average degree is
〈k〉 sim tα
and the aging eects can be neglected On the other hand for t ta the average
degree depends only on the aging time ta
〈k〉tat sim tαminus1
The prediction of NoPAD model are compatible with empirical data gathered from
the scientic collaboration network in PRL Society [107]
2414 Burstiness and memory Considering both the burstiness and the mem-
ory [23] processes the generative model of the network is the following
(1) The rst activation time τi for each node is extracted from the Ψ(τ ξi)
before starting the network evolution
(2) The time t is set on the node j with the smallest activation time t = τj
(3) The site j then contacts a new node with probability pj(kj) or a site in
its history of contacts with probability 1 minus pj(kj) In the last case the
integrated degree ki of all the nodes i remains the same
(4) A waiting time wj is drawn from Ψ(w ξi) and τj is updated to wj + τj
24 ACTIVITY-DRIVEN NETWORK 41
(5) Return to step 2
To obtain analytical results Ubaldi et al considered the approximation in which
individuals can only contact other nodes and never be contacted In this case they
consider the evolution of a single agent 0 with its waiting time distribution Ψ(w ξ0)
and memory function p(k) and study the master equation for the probabilityQ(k t)
that the individual makes a connection at time t and after that has degree k The
P (k t) then is obtained integrating over the time and the waiting time
P (k t) =
int t
0
dtprimeQ(k tminus tprime)int infintprime
dwΨ(w ξ0)
In general the results for the P (k t) depends on the average inter-event time of
Ψ(w ξ0) and on the asymptotic behavior of Ψ(w ξ0) for large w In particular
there are three intervals of interest for the exponent α that leads to dierent P (k t)
results [23]
P (k t)
1
(tw0)α
1+βfαβ
(Aprimeαβ
k
(tw0)α
1+β
)α lt 1
1
(tw0)1αminus α
1+βfαβ
(Aprimeαβ
kminusv(tw0)1
(1+β)
(tw0)1αminus α
1+β
)1 lt α lt 2β+2
β+1
1
(tw0)1
2(1+β)exp
minusAprimeβ(kminusCβ(tw0)
1(1+β)
)2
(tw0)1
1+β
α gt 2β+2β+1
where fαβ is a non-Gaussian scaling function v is the drift velocity of the peak of
the distribution Aαβ Aβ and Cβ are constant depending on the parameters β and
α
The average degree then can be written as
k(t) =
tα
1+β α lt 1
t1
1+β α gt 1
The equation for the degree distribution can be evaluated at xed time considering
ρ(k) =
intF (ai)P (ai k t)dai
When the activity distribution has a power-law decay the degree distribution is
given by
ρ(k) sim
kminus( 1+β
α (νminus1)+1) α lt 1
kminus((1+β)νminusβ) α gt 1
2415 Attractivness ADA A further extension of the simple activity-driven
networks was introduced by Pozzana et al [98] In their model they include the
characteristic of social systems to distinguish between more or less popular indi-
vidual The main idea is that a node i might be more popular then the others
introducing the concept of attractivness bi In this framework when a node i is
active it will target a node j with a probability depending on the js attractiveness
bj
24 ACTIVITY-DRIVEN NETWORK 42
Figure 242 Schematic representation of the model Straightlines represent the contact in the same community arch representthe connections between communities The active nodes are col-ored in red [77]
The distributions of the activity F (a) and of the acttractiveness G(b) can be either
uncorrelated or correlated aecting dierently the dynamical processes running on
the network
We will see in the next chapter how this model can be used to study epidemic
processes
Modular activity driven Datasets analysis stressed out the organization of real
networks in communities or modules where the density of connection is much
larger than the density of links between communities To include this feature in
the activity driven model Nadini et al [77] considered a network with N nodes and
tunable modularity where the size s of the communities is drawn from a given
distribution P (s) The heterogeneity of the modules size grasped from real data
suggested a heavy-tailed form for P (s) sim sminusω with s isin [sminradicN ]
In this framework each node is progressively assigned to a module of size s extracted
from P (s) and the generative process of the ADM network is the following
bull At each time the graph starts with N disconnected nodes
bull Each node is activated with probability ai∆t and creates m links (m can
be set to one)
bull Each link connects randomly within the community with probability micro or
outside the community with probability microprime = 1minus microbull At t+ ∆t all links are deleted
Where ∆t is the constant duration of the interactions which can be set to one
They considered the master equation for the probabilities Pc(s kc) and Po(s ko)
which are respectively the probability for a node of activity ai to belong to a
community of size s and have in-degree kc or out-degree ko respect to the community
at time t In the limit of large time t 1 and large degree k 1 they found the
24 ACTIVITY-DRIVEN NETWORK 43
analytic solution for the master equation for both the probability distributions
Pc(s kc) prop
exp[minus (kcminusmicro(aminus〈a〉)t)2
2micro(a+〈a〉)t
]t τc(s)
δ(kc minus (sminus 1)) t τc(s)
Po(s ko) prop exp[minus (kominusmicroprime(aminus〈a〉)t)2
2microprime(a+〈a〉)t
]forallt
Notice that while the in-community probability Pc depends on the size of the
community the out-community probability Po doesnt
Considering that kc + ko = k they determined the total probability distribution as
P (s k) =
int k
0
Pc(s kc)P0(k minus kc)dkc
In this framework they study the evolution of the average in-community degree of
each node given by
kc(a s t) = (sminus 1)
[1minus exp
(minus t
τ(a s)
)]where τ(a s) is the characteristic time that it takes for the degree kc(a s t) to
become maximal ie kc(a s t) sim s minus 1 On the other hand the out-community
average degree is given by
ko(a t) = microprime(a+ 〈a〉)
The total average degree is then given by
k(a s t) =
(a+ 〈a〉)t t τ(a s)
microprime(a+ 〈a〉)t+ sminus 1 t sim τ(a s)
microprime(a+ 〈a〉)t t τ(a s)
The long time evolution of the degree is linear in time hence for power-law activity
distribution F (a) = aminusν they obtained power laws degree distribution ρ(k) with
the same exponent ν
242 Random walks on activity-driven model The study of random
walks in TVN is a core concept for both analytical and computational models of
many real-world dynamical processes that mostly evolve on temporal scale-free
networks To study this formalism on the activity-driven network [96] we introduce
the propagator Π∆tirarrj of the random walk as the probability that a walker moves
from the node i to the node j in the time interval ∆t then we can write the master
equation for the probability Pi(t) that the walker is in the node i at time t
Pi(t+ ∆t) = Pi(t)
1minussumj 6=i
Π∆tirarrj
+sumj 6=i
Pi(t)Π∆tirarrj
Considering only the rst order terms in ∆t the expression for the propagator is
Π∆tirarrj
∆t
N(ai +maj)
24 ACTIVITY-DRIVEN NETWORK 44
where m as usual is the number of links red by an active nodes at each time step
For the activity-driven framework as we will see in detail later it is sometimes con-
venient to consider groups of the same activity class a assuming that they are statis-
tically equivalent in the limit ofN rarrinfin If we deneWa(t) = [NF (a)]minus1Wsumiisina Pi(t)
as the number of walkers in the same activity class a at time t we can write in the
continuous time limit ∆trarr 0 the dynamical equation for this quantity
parttWa(t) = minusaWa(t) + amw minusm 〈a〉Wa(t) +
intaprimeWaprime(t)F (aprime)daprime
where w is the average density of walkers per node The rst two terms account
for the active nodes which release all the walkers they have and are visited by the
walkers traveling from all the other nodes The last two terms account for the
contribution of the inactive nodes due to the activity of the nodes in all the other
classes We are interested in the stationary state in the innite time limit which
gives
Wa =amw + φ
a+m 〈a〉where φ =
intaprimeWaprime(t)F (aprime)daprime is the average number of walkers escaping from the
active nodes and it is constant in the stationary case Hence the problem reduces
to nd the solutions of the self-consistency equation
φ =
intaF (a)
amw + φ
a+m 〈a〉da
The result depends on the node activity and tends to a constant as a grows
In case of a heavy-tailed distribution the explicit solution for φ can be written in
term of the hypergeometric function We can also analyze the mean rst passage
time Ti or the average time needed for a walker to arrive to a vertex i starting
from any other node in a network which is given by
Ti =NW
maiW +sumj ajWj
CHAPTER 3
Epidemic Models
31 Introduction
Infectious diseases create a signicant problem for health and economic all around
the world The appearing of new diseases and the persistence of old ones make
epidemics modeling a fundamental tool to study this phenomenon and guide the
health policy around the world
Dierent approaches from dierent scientic elds have been used during the last
two centuries to describe epidemics from the Bernoulli model of the 1766 up to
now ranging from biology to computer science and mathematics [8 58]
The standard mathematical approach to epidemic processes is the compartmental
model [34 56 57] In this case the population is divided into classes or com-
partment depending on the stage of the disease It is possible to dene a va-
riety of compartments but for our purpose we will focus just on three of them
the susceptible stage (S) in which the individual can be infected the infectious
stage (I) in which the individual is infected and the recovered stage (R) in which
the individual is cured and immune to a reinfection In this work we will focus
on two compartmental models the Susceptible-Infected-Susceptible (SIS) and the
Susceptible-Infected-Recovered (SIR)
The main objective in the studying of an epidemic is to establish and formalize
the transitions between compartments so that it is possible to track the number of
individuals in each stage In this chapter we will consider some of the most relevant
and simplest models formulated up to now [92]
In the rst section we will introduce the classical mathematical approach to epi-
demics and dene some of the fundamental parameter needed to characterize this
phenomenon
In the second section we will use static networks concepts to understand how in-
troducing the topology aects the spreading process and in the last section we will
see the role of the time evolution of the network
32 Traditional models
The traditional approach studies the epidemics by using the mean-eld approxima-
tion [48] without introducing networks at all In this framework each individual
45
32 TRADITIONAL MODELS 46
interact with the whole population randomly Under this approximation the den-
sity of individuals Nσ in the compartment σ or its density ρσ = NσN fully
describes the state of the epidemics where σ can be S I or R in our case and N
is the total population
In the simplest denition of epidemics dynamics N is xed and all the other demo-
graphic processes can be ignored There are two types of transitions between the
compartments which completely dene the epidemic evolution the infection and
the recovery processes The recovery transition is spontaneous after a certain time
In the discrete time models an infected individual has a probability micro to recover at
any time step and the time it will spend in the infectious compartment will be microminus1
In the continuous time formulation it is generally assumed a Poisson process [32]
where now micro is a probability per unit time (rate) and we can dene the probability
that infected individuals remain in this state for a time τ as Pinf (τ) = microeminusmicroτ with
average infection time 〈τ〉 = microminus1 This means that the epidemic model can be
formulated in terms of a Markov process [52 126]
The infection transition occurs only if there is an interaction between a susceptible
and an infected individual and hence depends on the interaction pattern consid-
ered in the model and on several other factors Without information about the
connection between individuals the individuals are considered in the homogeneous
mixing approximation hence randomly interacting among each others In this case
the larger is the number of infectious agents among an individuals neighbors the
higher is the probability of the infection This naturally leads to the introduction
of the force of infection α which is the probability that an individual can contract
the infection in a single time step and in the continuous time limit is dened as the
rate
α = λρI
where λ depends on the specic disease and contact pattern of the population In
some cases λ can be split in the rate of infection per eective contacts λ and the
number of contacts k with other individuals
This approach can also be used considering the epidemic as a stochastic reaction-
diusion process where the individuals of each compartment can be seen as dierent
kinds of particles evolving according to specic interaction dened by the reaction
rate This framework is generally more complicated and goes beyond the objectives
of this introduction
We will present the classical results for epidemic processes considering the dynamics
in terms of deterministic ordinary dierential equations obtained applying the laws
of mass action in the mean-eld approximation In this case the change of the
density of the population in each compartment due to the interactions is given by
the force of infection times the average population density
32 TRADITIONAL MODELS 47
Notice that the mass-action approximation is not realistic In fact people interact
with a small fraction of the entire population and not randomly which underline the
importance of the introduction of a set of rules that dene an interaction structure
in the system Nonetheless the classical approach is useful to explore the core
mathematical features of the epidemic spreading
321 SIS Many real diseases dont confer immunity after the recovery which
let an individual susceptible to reinfection The simplest model that describes this
behavior is the SIS model in which only two states are possible the infected I and
the susceptible S The dynamics of this system can be described by the reaction
scheme
I + Sλminusrarr 2I I
microminusrarr S
where λ is the infection rate and micro is the recovery rate
The deterministic dierential equation describing the process is given by
parttρS = microρI minus λρIρS
parttρI = λρIρS minus microρI
Considering that ρI + ρS = 1 for a xed number of the total population the set of
equation can be simplied to
parttρI = (λminus microminus λρI)ρI
of which the solution is
ρI(t) =(
1minus micro
λ
) Ce(λminusmicro)t
1minus Ce(λminusmicro)tprime
where the integration constant is determined by the initial number of infected in-
dividuals ρ0
C =λρ0
λminus microminus λρ0
In the limit of large population small numbers of infected agents ρ0 rarr 0 and
C = λρ0(λminus micro) leading to
ρI(t) = ρ0(λminus micro)e(λminusmicro)t
λminus micro+ λρ0e(λminusmicro)tprime
If λ gt micro the population can never be totally infected and in the long-time limit the
stable state corresponds to a steady fraction of the population always infected with
the disease This fraction can be obtained imposing parttρI = 0 to give ρI = (λminusmicro)micro
which is called endemic state On the other hand when λ approaches to micro the
fraction of infected nodes in the endemic state goes to zero while if λ lt micro the
disease will die out exponentially
A fundamental parameter to evaluate the rising of an epidemic outbreak is the
basic reproduction number R0 [6] Consider a susceptible individual who catches
the disease in the early stage of an outbreak then R0 is dened as the average
number of additional infections caused by this agent before it recovers
32 TRADITIONAL MODELS 48
If R0 lt 1 the relative size of the epidemics vanishes because a single individual
cant generate enough secondary infection to sustain the spreading On the other
hand if R0 gt 1 the average fraction of infected agents grows exponentially The last
condition while necessary and sucient for deterministic models is only necessary
for stochastic models where uctuations of the number of infected individuals can
lead to the extinction of the infection for a small initial number of infected agents
The point R0 = 1 separates the two opposite behaviors dening the epidemic
threshold
In the SIS model the transition between epidemic and non-epidemic regime happens
at the point λ = micro also called epidemic transition point and the basic reproduction
number is given by R0 = λmicro
322 SIR For many diseases people retain their immunity after the recovery
process preventing them from a reinfection The simplest model to describe this
behavior is the SIR model In this framework a susceptible individual (S) can catch
the disease from an infected individual (I) which after a certain time can recover
and be removed from the dynamics (R) The dynamical process can be described
by the reaction scheme
I + Sλminusrarr 2I I
microminusrarr R
where contacts with infected individuals happen with an average rate λ while the
recovery process happens with a constant average rate micro
It is possible to dene the probability to recover in a time interval δτ as microδτ and
obtain the probability to stay infected after a total time τ as
limδτrarr0
(1minus microτ)τδτ = eminusλτ
The probability that an infected individual recover in the interval [δτ τ + δτ ] is
p(τ)dτ = microeminusmicroτdτ which is a standard exponential distribution meaning that the
recovery process is most likely to happen just after the infection takes place In
most cases this is quite unrealistic considering that people may remain infected
for much longer time depending on the disease We will see how this estimation
improves introducing the network
In terms of the fraction of individuals in each compartment the system is described
by the dierential equations
parttρS = minusλρIρS
parttρI = λρIρS minus microρI
parttρR = microρI
For a xed number of population we can consider the normalization condition
ρI + ρS + ρR = 1 the set of equations can be simplied eliminating the ρI variable
32 TRADITIONAL MODELS 49
1
09
08
07
06
05
04
03
02
01
Figure 321 SIR epidemic processes Density of nodes in eachcompartment depending on time t
obtaining
ρS = ρS0 eminusλρRmicro
where ρS0 is the fraction of susceptible individuals at time t = 0 and then using the
normalization condition it is possible to obtain
(321) parttρR = micro(1minus ρR minus ρS0 eminusλρ
Rmicro)
The solution can be written as
t =1
micro
int ρR
0
dx
1minus xminus ρS0 eminusλxmicro
which can not be evaluated in closed form but just numerically
As shown in gure (321) the fraction of susceptible individuals in the population
decreases monotonically and the fraction of recovered individuals increases mono-
tonically The fraction of infected goes up at rst as people get infected then down
again as they recover and eventually goes to zero when trarrinfin
On the other hand the fraction of susceptible individuals doesnt go to zero because
when ρI rarr 0 it is not possible to have new infections Also the fraction of recovered
doesnt reach one as trarrinfin and its asymptotic value represents the total number
of individuals that caught the disease hence is the total size of the outbreak which
can be useful to characterize the epidemic This can be calculated from the eq 321
imposing parttρR = 0 which gives ρR = 1minus ρS0 eminusλρRmicro
The most common choice for the initial condition is to consider the infection to
start either from a single individual or from a small fraction r of the population
In this case the initial values of the variables are ρR0 = 0 ρS0 = 1 minus rN and
ρI0 = rN so that in the limit of large population N rarrinfin the total outbreak size
is
ρRinfin = 1minus eminusλρRinfinmicro
These results indicate that the size of the epidemic continuously goes to zero for
λ le micro which means that the infected individuals recover faster than the susceptible
ones become infected so that the disease dies out
33 EPIDEMICS ON STATIC NETWORKS 50
λλc
ρNo epidemic
(absorbing phase)
Epidemic(acve phase)
Figure 322 Phase diagram of a SIS-like absorbing state phase transition
An individual that remains infected for a time τ by the same amount of time will
have contacted a number λτ of other individuals By denition the reproduction
number R0 is the average number of additional people that an infected individual
passes the disease to before they recover which is
R0 = λmicro
int infin0
τeminusmicroτdτ =λ
micro
As for the SIS model also in the SIR model the epidemic threshold falls in the point
where λ =micro in the long time regime
323 Epidemics and phase transition Epidemic processes are a typical
example of critical phenomena [133 46 65] In this case the phase transition is
between the non-epidemic (absorbing) and epidemic (active) phases characterized
by the order parameter ρσ and the control parameter λ
In the SIS case the order parameter is the density of the infected individuals ρI
determining the distinction between the non-epidemic and epidemic phases This
model belongs to the universality class of direct percolation which is the paradigm
of dynamical phase transitions
In the SIR model the order parameter is the size of the outbreak hence the density
ρR of all the population ever being infected
For both the SIS and SIR problem the control parameter is the infection rate λ
The critical point λc such that ρ = 0 for λ lt λc and ρ gt 0 for λ gt λc denes the
epidemic threshold of the system The phase diagram can be expressed in terms of
ρ(λ) as shown in the gure 322
33 Epidemics on static networks
Classical models of epidemic spreading consider the population to be fully connected
and the individuals to randomly interact within each other this assumption is
clearly unrealistic In general people have a regular set of acquaintances friends
and coworkers whom they interact with while ignoring the rest of the population
The potential contacts of an individual form a set that can be easily represented
33 EPIDEMICS ON STATIC NETWORKS 51
as a network As we will see from now on the network structure and its evolution
strongly aect the spreading of a disease
Introducing the network with N node and considering χ number of compartments
representing the stages of an epidemic process the state of the node i at time t
is given by the random variable Xi(t) where Xi(t) = σ means that the node i
belongs to the compartment σ at time t Considering the transitions between the
compartments as independent Poisson processes with certain rates the epidemic
process can be studied in terms of a Markov chain [52] At this point it is possible
to study the evolution of the probability for Xi(t) to be in a state σi isin [0 χ] at
time t
The other possible approach is to describe the evolution in terms of the master
equation (see 142) for the probability P (sσ t) to be in the compartment σ at the
time t where sσ is the set of states sσi (t) indicating that the node i belongs to the
compartment σ at time t
331 Individual based mean eld In the individual-based mean-eld (IBMF)
model the evolution equation is written in terms of the probability ρηi that node i
is in the state η for each node assuming that the dynamical state of each node is
statistical independent from the ones of its nearest neighbors ie that the probabil-
ity for a node i to be in a state η and for its neighbor j to be in a state ηprime is ρηi ρηprime
j
[70 43]
This approach keeps the full structure of the networks while using the mean eld
approximation to neglect the correlations between neighbors As a consequence
the solutions depend in general on the spectral properties of the adjacency matrix
they fail to describe the system when either the variable are highly correlated or
when the densities in a compartment are very small
3311 SIS IBMF The SIS epidemic process on a network can be described
by a Bernoulli random variable Xi(t) isin 0 1 where Xi = 0 corresponds to the
susceptible state and Xi(t) = 1 corresponds to the infected state of the node i at
time t [70 125] Hence the probability for a node i to be infected at time t is
given by ρIi (t) = Pr[Xi(t) = 1] which for a Bernoulli variable corresponds to the
expectation value E[Xi(t)] while the probability to be susceptible is 1minusρIi (t) Thegeneral exact equation that describes the expectation of being infected for each
node i is given by
(331) parttE[Xi(t)] = E
minusmicroXi(t) + [1minusXi(t)]λ
Nsumj=1
aijXj(t)
where the second term is the expectation value that the node i recovers with rate
micro and being susceptible [1minusXi(t)] is infected by its neighbors In this case aijare the elements of the adjacency matrix This formalism can be extended to both
time dependent adjacency matrix A(t) and asymmetric adjacency matrix From the
33 EPIDEMICS ON STATIC NETWORKS 52
formula above we can say that the time evolution of the probability to be infected
is aected by two mechanisms if the node is infected then parttE[Xi(t)] decreases
with a rate micro while if it is healthy it can be infected with rate λ
For static networks Eq (331) reduces to the following [105]
(332) parttρIi (t) = minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)minus λ
Nsumj=1
aijE[Xi(t)Xj(t)]
Now we can apply the IBMF approximation to close the equation assuming the
statistical independence of the states of two neighboring nodes ie
E[Xi(t)Xj(t)] = E[Xi(t)]E[Xj(t)] = ρIi (t)ρIj (t)
and Eq (332) becomes
(333) parttρIi (t) = minusmicroρIi (t) + λ
[1minus ρIi (t)
] Nsumj=1
aijρIj (t)
Hence the time evolution of the probability ρIi (t) is given by minus the probability
that an infected node recovers with rate micro plus the probability that a susceptible
node gets the infection from one of its nearest infected neighbors with rate λ
The epidemic threshold is obtained applying the linear stability analysis to (333)
and studying the eigenvalues problem for the Jacobian J with elements Jij =
minusδij + λmicroaij We are in the endemic case when the largest eigenvalues Λ1 of J is
positive [70]which leads to the condition
λ ge λIBMFc =
1
Λ1
For heterogeneous networks with power-law degree distribution P (k) sim kminusγ Λ1 simmax
radickmax
langk2rang 〈k〉
[30] where kmax is the maximum degree of the network
the epidemic threshold is
λIBMFc sim
1radickmax γ gt 52
〈k〉 langk2rang
2 lt γ lt 52
This means that for every network where the maximum degree is a growing function
of the network size the epidemic threshold vanishes in the thermodynamic limit
3312 SIR IBMF Also in the SIR case the approach of the IBMF approxi-
mation is to write the full master equation for the probabilities of the states for each
node and to obtain from it the deterministic evolution equation for the quantities
parttρSi (t) = minusλ
Nsumj=1
aij 〈SiIj〉
parttρIi (t) = λ
Nsumj=1
aij 〈SiIj〉 minus microρIi (t)
33 EPIDEMICS ON STATIC NETWORKS 53
where Si and Ii are Bernoulli variable such that ρSi = 〈S〉i and ρIi = 〈I〉i are the
probability for the node i to be susceptible or infected respectively while 〈SiIj〉 isthe join probability of being in the state SiIj
The IBMF approximation 〈SiIj〉=〈S〉i 〈I〉j introduced to close the equation bringsout a physical interpretation problem
parttρSi (t) = minusλ
Nsumj=1
aijρSi ρ
Ij
parttρIi (t) = λ
Nsumj=1
aijρSi ρ
Ij minus microρIi (t)
As shown by Sharkey [105] this approximation is equivalent to write the evolution
equation of 〈SiIj〉 implying that a node can be susceptible and infected at the
same time In this case the IBMF approximation leads to the same result for the
epidemic threshold of the SIS case giving λc = 1Λ1 For heterogeneous power-law
distributed networks and γ gt 3 this result leads to a vanishing epidemic threshold
which is not correct as shown in [28]
332 Degree based mean eld In the degree-based mean eld (DBMF)
or heterogeneous mean eld (HMF) approximation all the nodes of the same degree
k are considered statistically equivalent which means that any node with degree
k is connected with probability P (k|kprime) to every node of degree kprime In this case
the relevant variables specify the degree class of a node [91] and the dynamical
equation is written in terms of the probability ρηk(t) that a node with degree k
is in the compartment η at time t The variables which are not independent
must satisfy the conditionsumη ρ
ηk(t) = 1 while the total fraction of agents in the
same compartment η is given by ρη(t) =sumk P (k)ρηk(t) where P (k) is the degree
distribution
In this framework the network itself is considered in a mean eld perspective
preserving only the degree information and the correlation between two nodes [35]
In this sense the DBMF is equivalent to use the ensemble average of the adjacency
matrix in the IBMF theory
aij =kjP (ki|kj)NP (ki)
which represents the probability that the nodes i and j are connected this is also
referred to annealed network approximation
The solutions of DBMF models generally depend on the statistical properties of the
networks but despite being a strong approximation it well describes spreading
processes evolving on networks for which the interactions changes on time scales
much faster than the dynamics on them
33 EPIDEMICS ON STATIC NETWORKS 54
3321 SIS DBMF In the DBMF approximation the dynamical equation for
the SIS process can be described by the law of mass action [91]
(334) parttρIk (t) = minusmicroρIk(t) + λk
[1minus ρIk (t)
]sumkprime
P (kprime|k) ρIkprime(t)
The rst term considers the recovered nodes of degree k The second accounts for
the infection of new nodes considering the probability that a node vk of degree k
is susceptible 1minus ρIk(t) times the infection rate λ and probability that a node vkprime
of degree kprime is infected and connected to vk with probability P (kprime|k) summed over
all possible values of kprime We can divide Eq 334 by micro and imposing it equal to one
without loss of generality The epidemic threshold is obtained studying the linear
stability of the equation and performing a rst order expansion in ρIk(t)
parttρIk (t)
sumk
JkkprimeρIkprime (t)
where Jkkprime = minusδkkprime+λkP (kprime|k) is the Jacobian matrix element The healthy phase
corresponds to a null steady state which is stable when the largest eigenvalue of
the Jacobian ΛM is negative On the other the solution ρk = 0 is unstable if exist
at list on positive eigenvalue of the Jacobian matrix this means that the epidemic
phase emerges when λΛM minus 1 gt 0 ie when
(335) λ gt λDBMFc = Λminus1
M
as shown in Ref [19]
For uncorrelated networks
(336) P (kprime|k) = kprimeP (k) 〈k〉
and Eq 334 can be written as
(337) parttρIk (t) = minusρIk(t) + λk
[1minus ρIk (t)
]Θ(λ)
where
Θ(λ) =sumk
kP (k)
〈k〉ρIk(t)
is the probability that a random chosen link leads to an infected node
From the stationary condition it is possible to obtain an expression for the proba-
bility ρIk (t) given by
ρIk (t) =λkΘ(λ)
1 + λkΘ(λ)
This indicates that for uncorrelated networks the higher the nodes degree is the
higher is its probability to be infected implicating that high heterogeneity in the
connectivity patterns strongly aects the spreading of a disease
Notice that Θ(λ) can be computed solving the self-consistency equation
(338) Θ(λ) =sumk
kP (k)
〈k〉λkΘ(λ)
1 + λkΘ(λ)
33 EPIDEMICS ON STATIC NETWORKS 55
In this case the epidemic threshold can be derived either substituting the ex-
pression 336 in the Jacobian and computing the eigenvalue or imposing that the
self-consistency equation 338 admits a non-zero solution obtaining
(339) λ gt λDBMFuncc =
〈k〉〈k2〉
The critical behavior of the order parameter around the critical point can be ob-
tained from 338 giving ρIk (t) sim (λminus λDBMFc )η
DBMFSIS where ηDBMF
SIS is the critical
exponent Moreover for networks with power-law degree distribution P (k) sim kminusγ
with exponent 2 lt γ le 3 in the limit of innite scale networks the epidemic
threshold tends to zero while the critical exponent is larger than 1 This means
that while the disease spreads more easily the epidemic activity grows very slowly
increasing the spreading rates making the epidemic less threatening
Notice that for regular networkslangk2rang
= 〈k〉2 recovering the result λDBMFc = 1 〈k〉
3322 SIR DBMF To extend the DBMF approximation to the SIR model
it is necessary to introduce also the partial densities of the recovered and of the
susceptible nodes with degree k ρRk (t) and ρSk (t) respectively which fulll the nor-
malization condition ρRk (t) + ρSk (t) + ρIk (t) = 1 for nite size population The set
of the equations describing the process is given by
parttρIk (t) = minusρIk(t) + λkρSk (t)
sumkprime
P (kprime|k) ρIkprime(t)(3310)
parttρRk (t) = ρIk(t)
Also in this case as well the linear stability analysis leads to the value for the
epidemic threshold which is the inverse of the largest eigenvalue of the adjacency
matrix
For the SIR model the order parameter is the number of recovered individuals at
the end of the epidemics ie ρRinfin(t) = limtrarrinfinsumk P (k)ρRk (t)
For uncorrelated networks it is possible to integrate the rate equation over time to
study the whole temporal evolution of the process introducing the function
φ(t) =sumk
kP (k)
〈k〉ρRk (t)
In general the solution depends on the dierential equation for the function φ(t)
but in the limit of innite time it is possible to obtain the nal prevalence
ρRinfin =sumk
kP (k)(1minus eminusλkφinfin
)where
(3311) φinfin = 1minus 1
〈k〉minussumk
kP (k)
〈k〉eminusλkφinfin
33 EPIDEMICS ON STATIC NETWORKS 56
The epidemic threshold can be obtained from 3311 giving λc = 〈k〉〈k2〉 Moreover
for power-law degree distributed networks with P (k) sim kminusγ the equation for the
order parameter is ρRinfin sim (λminus λc)ηSIR [76]
Notice that in case of annealed networks the results above are exactbut in case
of static networks it is possible to improve the models prediction considering that
in the SIR model the reinfection of a recovered node is prohibited and the disease
cant propagate through the neighbors that have already been infected The eect
on this approximation can be included modifying the sum in the second rhs term
of the 3311 P (kprime|k)rarr P (kprime|k) (kprime minus 1)kprime giving a new largest eigenvalue of the
adjacency matrix
Λ1 =
langk2rang
〈k〉minus 1
which corresponds to the epidemic threshold
λc =〈k〉
〈k2〉 minus 〈k〉
An important insight in particular for the SIR-like models is the time scale evo-
lution of an epidemic outbreak which is of order (λΛ1)minus1and in this case is given
by
τ =〈k〉
λ 〈k2〉 minus (micro+ λ) 〈k〉Notice that as for the epidemic threshold the time-scale of an epidemic outbreak
vanishes when the second moment of the degree distribution diverges for example
in scale-free networks This mechanism can be extensively studied in a scale-free
network with computer simulations showing that at the beginning the infection
reaches the hubs and invades the rest of the networks via a cascade process [15 16]
333 Other Results
3331 SIS A relevant result for the SIS model provides a lower bound for
the epidemic threshold This was introduced by Mieghem [69] considering the
inequality 0 lesumNj=1 ajiXi(t)Xj(t) where Xj(t) are the Bernoulli random variable
introduced in the eq 331 In this case it is possible to write
parttρIi (t) le minusmicroρIi (t) + λ
Nsumj=1
aijρIj (t)
Considering the vector W = (ρI1 ρIN ) in a network of N nodes the solution of
the inequality is
W (t) le e(λmicroAminus1)tW (0)
The inequality is dominated by the term λmicroΛ1minus1 where Λ1 is the largest eigenvalue
of the adjacency matrix A When λmicroΛ1 minus 1 le 0 Wi(t) = ρIi (t) tends to 0 and the
fraction of infected individuals rapidly decreases ending the epidemic spreading
33 EPIDEMICS ON STATIC NETWORKS 57
This imposes a lower bound for the epidemic threshold
λc ge1
Λ1
which is the same result as for the IBMF model
3332 SIR The SIR process in the long time regime can be mapped to a
bond percolation problem [81] In this framework the links in a network are kept
with probability 1 minus p and removed with probability p The probability that a
randomly chosen link doesnt attach to a vertex connected to a giant component is
given by
(3312) u = 1minus p+sumk
kP (k)
〈k〉(1minus p+ pu)kminus1
which is the equation for degree uncorrelated networks with no loops in which a
randomly chosen edge points to a node of degree k with probability kP (k) 〈k〉The probability that a randomly chosen node belongs to the giant component is
(3313) PG(p) = 1minussumk
P (k)(1minus p+ pu)k
Introducing the degree distribution generating function G0(z) =sumk P (k)zk and
the excess degree generating function G0(z) =sumk(k + 1)P (k + 1)zk 〈k〉 it is
possible to write the equations 3313 and 3312 as
u = 1minus p+G1(1minus p+ pu)
PG(p) = 1minusG0(1minus p+ pu)
The condition for the existence of a giant component translates into the condition
for the existence of a nonzero solution which is
p gt pc =Gprime0(1)
Gprimeprime0(1)=
〈k〉〈k2〉 minus 〈k〉
The behavior of the order parameter can be found performing the expansion of the
generating function near the critical point around the nonzero solution obtaining
PG(p) sim (pminuspc)βperc where the critical exponent in case of homogeneous networks
is βperc = 1 For heterogeneous networks with degree distribution P (k) sim kminusγ in
the thermodynamic limit N rarrinfin the percolation threshold tends to zero for γ lt 3
and the critical exponents take the values
βperc =
1
(3minusγ) for γ lt 3
1(γminus3) for 3 lt γ le 4
1 for γ gt 3
As shown in [78] the probability that a link exists p is related to the probability
that an infected node can transmit the disease to a connected susceptible node
Lets consider the SIR model with uniform infection time τ ie the recovery time
after the infection and infection rate λ the transmissibility T is dened as the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 58
probability that an infected node transmits the disease to a susceptible node and
in the continuous time limit it is given by
T = 1minus limδtrarr0
(1minus λδt)τδt = 1minus eminusτλ
Now the cluster of the bond percolation problem to which the initial node belongs
is the same as the set of recovery nodes of an SIR outbreak generated from a single
node where the occupation probability p is now T The correspondence is exact
and for tree-like networks we have
Tc =〈k〉
〈k2〉 minus 〈k〉
λc =1
τln
langk2rangminus 〈k〉
〈k2〉 minus 2 〈k〉where the behavior of the outbreak size close to the epidemic threshold is given
by the exponent βperc Notice that this means that the epidemic thresholds has
qualitatively dierent behavior for scale-free networks (γ lt 3) and scale-rich ones
(γ ge 3) For scale-free networks the threshold vanishes meaning that this type of
structures are extremely vulnerable to disease spreading
In presence of loops and multiple spreading paths the possible correlation can
invalidate the result However for random graphs which are locally treelike this
result still stands in the thermodynamic limit where the loops are innitely long
We can extend the result to non uniform infection times assuming that τi and λijvary between individuals In this case the trasmissibility Tij depends on the edge
(i j) and neglecting the uctuation we can replace Tij with its mean value
〈Tij〉 = 1minusintdτ
intdλeminusλτQ(λ)P (τ)
where P and Q are the distributions of τi and λij respectively For non-degenerate
infection times exponentially distributed and constant recovery rate performing
the integral we obtain the epidemic threshold
λc =〈k〉
〈k2〉 minus 2 〈k〉
Notice that this approximation fails when correlations are involved in fact if an
individual recovers quickly the probability that it will transmit the disease to its
neighbors is small while it is much higher if it recovers slowly When τi are degen-
erate and λij vary this approximation is not exact [71]
34 Epidemics on time evolving networks
In all the model of epidemic spreading presented until now we have supposed a
fundamental approximation the dynamics of the network doesnt aect the dy-
namics of the contagion process Indeed we have considered the limit of extreme
34 EPIDEMICS ON TIME EVOLVING NETWORKS 59
S
I
I
I I
S
S R
I
I
I I
Figure 341 SIR and SIS processes on the activity-driven network
time scale separation between the network and the contagion process either consid-
ering the network frozen with time scales much larger than the dynamical process
or considering the network rewiring on much faster time scale than the contagion
process
In reality especially in social dynamics the contacts change on the same time-scale
of spreading phenomena For instance we generally interact with a small number
of contacts at the time rather than with all our friends simultaneously hence the
time evolution of the networks strongly aect the spreading process dynamics in
real social systems
341 Simple activity-driven model (AD) As described in the previous
chapter the activity driven model is one of the most versatile models of TVN
that allows the analytical study of a dynamical process on the network In this
framework the time variable is embedded in the nodes activity a which represents
the probability for the node to form a number m of links at time t The activity is
drawn from the distribution F (a)
In the original formulation of the epidemic spreading on the ADN Perra et al [94]
considered the evolution of the number of infected individuals Ita in the class of
activity a at time t They used the HMF approximation considering all the nodes
with the same activity statistically equivalent
The dynamic process for both the SIS and the SIR model is explained in the gure
(341) while the equation for the SIS is given by
It+1a minus Ita = minusmicroIta + λm(Na minus Ita)a
intdaprime
ItaN
+ λm(Na minus Ita)
intdaprimeaprime
ItaN
where Na = NF (a) is the total number of individual in the activity class a micro and
λ are the recovery and infection probabilities respectively and (NaminusIta) = Sta is the
number of susceptible individual in the activity class a at time t The rst term on
rhs represents the recovery nodes the second accounts for the probability that a
susceptible node activates and becomes infected contacting an infected node while
34 EPIDEMICS ON TIME EVOLVING NETWORKS 60
the third is the probability that a susceptible node is infected when contacted by
one of its neighbors
Summing over all the activity classes and neglecting the second order correlations
in the continuous time limit the equation reduces to the system of equations
parttI = minusmicroI + λm 〈a〉 I + λmΘ
parttΘ = minusmicroΘ + λmlanga2rangI + λ 〈a〉Θ
where Θ =intdaIaa
Studying the linear stability analysis the system can be exactly solved and requiring
the largest eigenvalue of the Jacobian
J =
(minusmicro+ λm 〈a〉 λm
λmlanga2rang
minusmicro+ λm 〈a〉
)to be positive it is possible to obtain the epidemic threshold
λ
microge 1
m
〈a〉〈a〉+
radic〈a2〉
As we can see the threshold depends only on the activity distribution and in par-
ticular the uctuations of the activity tends to dump the threshold The same
result can be obtained for the SIR model We will introduce other variations of
the AD model and in particular we will discuss thoroughly what happens when we
introduce a memory process
342 Activity-driven model with attractiveness (ADA) The attrac-
tiveness b measures the probability for an individual to target for an interaction
distributed according to G(b) As we discussed the introduction of this process
in the activity driven model aect the spreading of a disease [97] In the general
case the system is described by the distribution H(a b) of both the activity and the
attractiveness Lets rst study the SIS process in the HMF approximation where
all the nodes with the same activity and attractiveness are statistically equivalent
Then we can write the equation for the number of infected nodes Itab in the class
of activity a and in the class of attractiveness b at time t in limit N 1 where N
is the total number of nodes
It+1ab minus I
tab = minusmicroItab +
λm
N 〈b〉(Nab minus Itab)
asumaprimebprime
Itaprimebprimebprime + b
sumaprimebprime
aprimeItaprimebprime
The main dierence respect to the AD case is that now the probability for a node
in the class (a b) to be contacted depends on b In the limit of Iab Nab where
Nab is the number of nodes in the class (a b) the equation reduces to the system
parttI = minusmicroI +λm
〈b〉[〈a〉Φ + 〈b〉Θ]
34 EPIDEMICS ON TIME EVOLVING NETWORKS 61
parttΘ = minusmicroΘ +λm
〈b〉[langa2rang
Φ + 〈ab〉Θ]
parttΦ = minusmicroΦ +λm
〈b〉[〈ab〉Φ +
langb2rang
Θ]
with eigenvalue l0 = minusmicro lplusmn = λm〈b〉
(〈ab〉 plusmn
radic〈a2〉 〈b2〉
)minusmicro Imposing the condition
of positive eigeinvalue for the outbreak to happen we obtain the epidemic threshold
λ
microge 1
m
〈a〉 〈b〉〈ab〉+
radic〈a2〉 〈b2〉
If the activity and the attractiveness are uncorrelated then H(a b) = F (a)G(b)
and the epidemic threshold become
1
m
1
1 +radic〈a2〉〈b2〉〈a〉2〈b〉2
We can also consider a deterministic correlation between a and b imposingH(a b) =
F (a)δ(bminus q(a)) where q(a) is a function determining b for a given a Considering
the relation G(b) = F (qminus1(b))|dqminus1(b)db| then if one of the variables is power-law
distributed so the other is
In the particular case of q(a) = aγc the epidemic threshold is given by
λ
microge 1
m
〈a〉 〈aγc〉〈a1+γc〉+
radic〈a2〉 〈a2γc〉
In both the choices for the distributionH(a b) we can notice that for heterogeneous
systems the epidemic threshold is lowered by the attractiveness This analytical
results are valid also for the SIR process and can also be veried by numerical
simulations
343 Activity-driven model with modularity In the previous chap-
ter we introduced a community structure of size s and distributed according a
P (s) sim sminusω on the AD model In this section we will describe the derivation of the
epidemic threshold for the SIS and SIR processes obtained by Nadini et al [77] in
this framework
To write the mean-eld equation governing the dynamic of the epidemic process
we need to introduce the activity block variable indicating for each compartment
the number of individuals with activity a and community size s at time t hence we
will have the block variable Sas(t) Ias(t) and Ras(t) The evolution for the SIR
process is governed then by the equation
parttIsa = minusγIas + λSas
[microaIss
+ (1minus micro)aI
N
]+ λ
sumaprime
[microIaprimes
Sass
+ (1minus micro)IaprimesSasN
]where Is is the number of infected individual in a community of size s while I is the
number of infected in the whole network Here γ is the recovery probability λ is the
34 EPIDEMICS ON TIME EVOLVING NETWORKS 62
infection probability and micro is the probability of joining a community The second
and the third terms on the rhs represent the probability that a susceptible node in
a community of size s contacts an infected node inside its community (Is) or outside
(I) while the fourth and the fth terms are the probability for an infected node of
activity class aprime to connect with a susceptible node inside or outside its community
respectively In the approximation of small community size ie N minus s sim N and
small initial numbers of infected individuals in each community (I minus Is sim I) it is
possible to write with the same considerations of the simple AD the set of dynamical
equations
parttI = minusγI + λ 〈a〉 I + λΘ + λmicrosums
(〈a〉s minus 〈a〉)Is
parttΘ = minusγΘ + λlanga2rangI + λ 〈a〉Θ + λmicro
sums
(langa2rangsminuslanga2rang)Is + (〈a〉s minus 〈a〉)Θs
where 〈an〉s are the moments of the activity distribution in any community of size
s Θ =suma aIa and Θs =
suma aIas
In the limit of microrarr 0 the model recovers the simple AD threshold for both the SIS
and SIR processes while for microrarr 1 numerical results for the threshold show that it
goes in opposite directions In the SIR model the reinfection process is not allowed
by the dynamics in this case strongly cohesive communities with high modularity
in the connectivity patterns once recovered tends to block the spreading On the
other hand in the SIS model the reinfection mechanism promotes the spreading
among the community and high modularity lower the epidemic threshold
344 Epidemics in continuous time evolving networks A general ap-
proach to calculate the epidemic threshold on temporal networks was proposed by
Valdano et al [124 123] In this framework the temporal network is described by
the time dependent adjacency matrix A(t) in the time interval t isin [0 T ] which
completely characterizes the epidemic threshold in the SIS case The rst approach
is to discretize the time in steps of length ∆t obtaining a discrete sequence of
adjacency matrices Ahh=1Tstep
The SIS dynamics is introduced by meaning of a discrete-time Markov chain equa-
tion for the probability phi that a node i is infected at the time step h given by
ph+1i = (1minus Phi)
1minusprodj
(1minus λ∆tAhijphj
+ phi(1minus micro∆t)
where λ∆t and micro∆t are the probability to be infected and to recover respectively
Now we can introduce the infection propagator
P (Tstep) =
Tstepprodk=1
[1 + λ∆tAk minus micro∆t]
where the generic element Pij(Tstep) represents the probability that the disease
propagates from a node i at time 1 to a node j at time Tstep when λ is close to
34 EPIDEMICS ON TIME EVOLVING NETWORKS 63
λc in the quenched mean-eld approximation Wang et al [131] and Goacutemez et al
[43] In the discrete time problem the epidemic threshold can be found solving the
equation
(341) ρ[P (Tstep)] = 1
for the spectral radius ρ In particular when the contagion dynamic is much faster
than the network evolution the adjacency matrix is a constant Ah = A and the
solution of the equation 341 gives the epidemic threshold λc = 1ρ(A) which is the
same result of the quenched case On the other hand in the annihilated case when
the dynamics is much slower than the network evolution temporal correlations are
lost and we need to consider the number of times edges were active during the
whole interval Tstep Considering A =sumhA
h the epidemic threshold in this case
is given by λc = Tstepρ(A)
The extension to the continuous time limit can be obtained considering the evolu-
tion equation for the infection propagator
˙P (t) = P (t)[minusmicro+ λA(t)]
Introducing the rescaled trasmissibility γ = λmicro it is then possible to solve this
equation in terms of series of micro Blanes et al [17]
P (t) = 1 +sumjgt0
microjP (j)(t)
where
P (j)(t) =
int t
0
dx1
int x1
0
dx2
int xjminus1
0
dxj [γA(xj)minus 1][γA(xjminus1)minus 1] middot middot middot [γA(x1)minus 1]
For t = T the epidemic threshold can be found solving the equation ρ[P (T )] = 1
In the special case of weak commutation[A(t)
int t0A(tprime)dtprime
] P (T ) = eT [minusmicro+λ〈A〉]
and the threshold is given by
λc = microρ(〈A〉)
The epidemic threshold of the AD model can be retrieved by this method using the
asymptotic form of the average adjacency matrix 〈A〉ij = (mδN)(ai + aj) of the
model where ai is the activity of the node i m is the number of links generated
by an active node and δ is the lasting duration of the contacts between nodes
CHAPTER 4
Epidemic Spreading and Aging in Temporal
Networks with Memory
41 Introduction
In this chapter we study the SIS and SIR epidemic processes on activity-driven
time-varying networks with memory We formulate the activity-based mean-eld
(ABMF) approach deriving analytically a prediction for the epidemic threshold as
a function of the activity distribution and of the parameter tuning the memory
The results show that memory overall reinforces the eects of activity uctuations
leading to a lower value for the epidemics threshold
The mean-eld approach provides exact results when the epidemics start after the
network has evolved for a long time In this regime the dynamical process is equiv-
alent to an epidemic model dened on an eective static network explained in the
dissertation We show however that strong aging eects are present and that in
the preasymptotic regime the epidemic threshold is deeply aected by the starting
time of the epidemics In particular due to memory at short times the dynamics
displays correlations among the infection probabilities of the nodes which have al-
ready been in contact with The correlations give rise to backtracking eects that
cannot be neglected In this case typically the threshold of the SIS and SIR models
are respectively smaller and larger than the mean-eld prediction We explain in
detail the origin of such deviations opening new perspectives for epidemic control
of disease and information spreading on temporal networks with high correlations
The chapter is organized as follows In Section 2 we will refresh the activity-driven
model for network topology in the presence of memory and how the SIS and SIR
dynamics evolve on top of it In Section 3 after a brief reminding of the analytical
approach to epidemic dynamics on memoryless activity-driven networks we will
describe in detail the eects of the introduction of the memory to this method
deriving predictions for the epidemic threshold In Section 4 we compare analytical
predictions with numerical results obtained by considering both an eective static
network and the full time-evolution of the topology The nal Section presents
some concluding remarks and perspectives for future work
64
42 THE MODEL 65
42 The model
421 Activity-Driven Networks with memory In activity-driven mod-
els each node vi (i = 1 N) of the graph Gt has an activity ai assigned randomly
according to a given distribution F (a) The dynamics occurs over discrete tempo-
ral steps of length ∆t At each step with probability ai∆t the vertex vi becomes
active and gets linked to m other vertices Connections last for a temporal interval
∆t At the next time step t + ∆t all existing edges are deleted and the proce-
dure is iterated Notice that the activity a is a probability per unit time Real
data observations indicate that human interactions are very often characterized by
skewed and long tailed activity distributions so F (a) is typically assumed to be a
power-law F (a) = Baminus(ν+1) with ε le ai le A Since in our simulations we will
keep the time interval ∆t = 1 the upper cuto is naturally set to A = 1
In order to consider the tendency of individuals to persist in their social connections
we can introduce a reinforcement mechanism The nodes are endowed with a
memory of their previous contacts and they contact preferably individuals belonging
to their social circle For an active node vi which has already contacted ki(t)
dierent nodes at time t this process is described by assuming that the node
connects with a new node with probability
(421) p[ki(t)] = [1 + ki(t)ci]minusβi
Complementary with probability 1 minus p[ki(t)] the node establishes a connection
with a previously contacted The parameter βi controls the memory process and
the constant ci sets an intrinsic value for the number of connections that node vi is
able to engage in before memory eects become relevant The probability depends
on the degree of the integrated network at time t ki(t) ie the number of nodes
that vi has contacted up to time t We will call Aij(t) the adjacency matrix of this
integrated network Empirical measures on several datasets are compatible with
constant values of βi and ci so we will consider ci = 1 and set βi = β independently
of the site i With this choice the function p(middot) is independent from i For now on
we will consider that the number of link m generated at each time-step is equal to
one with no loss of generality
As shown in [121] the asymptotic form of the degree distribution for the integrated
network can be derived analytically In particular in the regime 1 k N the
degree of nodes of activity a is narrowly distributed around the average value
(422) k(a t) = C(a)t1(1+β)
ie the degree of each node increases sublinearly in time with a prefactor depending
on its activity The prefactor C(a) is determined by the condition
(423)C(a)
1 + β=
a
Cβ(a)+
intdaF (a)a
Cβ(a)
43 ANALYTICAL RESULTS 66
In the memoryless case β = 0 where an active node connects always with a ran-
domly chosen vertex Eq (423) gives C(a) = a+ 〈a〉 recovering the result of [113]We will denote in general with 〈g〉 =
intdaF (a)g(a) the average of a function of the
activity g(a) over the network
422 The epidemic process We now turn to the spreading of infectious
diseases on activity-driven temporal networks with memory We start by consid-
ering the standard Susceptible-Infected-Susceptible (SIS) model introduced in the
previous chapter ie the simplest description of a disease not conferring immunity
An infected node can turn spontaneously susceptible with rate micro while an infected
node transmits the infection over an edge to a susceptible neighbor with rate λ
The two elementary events are therefore
(424) I + Sλminusrarr 2I I
microminusrarr S
In the Susceptible-Infected-Recovered (SIR) model the disease confers immunity
and the dynamics is described by the following reaction scheme
(425) I + Sλminusrarr 2I I
microminusrarr R
The epidemic process on activity-driven networks is implemented by iterating dis-
crete time steps of duration ∆t
bull at the beginning of each time step there are N disconnected vertices
bull with probability ai∆t a vertex vi becomes active and connects to a previ-
ously linked node with probability 1minus p(ki) or with a new node vj with
probability p(ki) in this second case ki(t) kj(t) and Aij(t) are increased
by one unit
bull if one of the nodes connected by the link is infected and the other one is
susceptible the susceptible becomes infected with probability λ
bull a vertex vj if infected becomes susceptible (SIS) or recovers (SIR) with
probability micro∆t
In activity-driven models λ is a pure number ie the probability that in a single
contact the infection is actually transmitted while micro is still the rate of recovery for
a single individual Ignoring the inhomogeneity in the activities one can estimate
the total rate for the infection process per node as λ 〈kprime〉 where 〈kprime〉 = 2 〈a〉 is theaverage degree per unit time this is the quantity to be compared with the recovery
rate per node micro
43 Analytical results
Epidemics on memoryless activity-driven networks The epidemic spread-
ing for the memoryless case β = 0 has been studied in adopting an ABMF approach
The epidemic state of a node when averaged over all possible dynamical evolutions
43 ANALYTICAL RESULTS 67
only depends on the value of its activity ai In particular one can dene the prob-
ability ρ(ai t) that a node with activity ai is infected at time t The corresponding
evolution equation is
(431) parttρ(ai) = minusmicroρ(ai) + λ[1minus ρ(ai)] + 1Nminus1
sumj 6=i
[aiρ(aj) + ajρ(aj)]
The rst term on the right side is due to recovery events the second term takes into
account the event that a susceptible node of class ai becomes active and contracts
the disease by connecting to an infected individual while the third term is the
analogous term for the case of a susceptible node that independently of her own
activity is contacted by an infected active individual
The description in terms of quantities that only depend on the activity is concep-
tually analogous to the heterogeneous-mean-eld approach for dynamical processes
on static networks [89] In that case one assumes that the only property deter-
mining the epidemic state of a node is the degree k and then derives equations
for the probabilities ρk An important dierence must however be stressed As-
suming the epidemic state to depend only on the degree is an approximation for
static networks because it neglects the quenched nature of the network structure
that makes properties of nodes with the same degree but embedded in dierent
local environments dierent In practice this assumption is equivalent to replacing
the actual adjacency matrix of the network (Aij equal to 0 or 1 depending on the
presence of the connection between vi and vj) with an annealed adjacency matrix
Pij = kikj(〈k〉N) [35] expressing the probability that vertices vi and vj with
degree ki and kj are connected The annealed approach is an approximation for
static networks while it is exact for networks where connections are continuously
reshued at each time step of the dynamics since the reshuing process destroys
local correlations Because in memoryless activity-driven networks connections are
extracted anew at each time step the ABMF approach provides exact results in
this case
Equation (431) can be analyzed by means of a linear stability analysis yielding
for large N the threshold [94]
(432)
(λ
micro
)ML
=1
〈a〉+radic〈a2〉
The same result can be derived for the SIR case
Epidemics on activity-driven networks with memory
Individual-based mean-eld approach In presence of memory interactions oc-
cur preferably with a subset of the other nodes (the social circle) creating correla-
tions Therefore we implement a dierent individual-based mean-eld approach
keeping explicitly track of the evolution of social contacts (ie of the memory) Let
us rst consider the SIS model The observable of interest is the probability ρi(t)
43 ANALYTICAL RESULTS 68
that node vi is infected at time t Its evolution can be written as
(433)
parttρi(t) = minusmicroρi(t)+
λ [1minus ρi(t)]
sumj ai [1minus p(ki)] Aij(t)ki
ρj(t) +sumjiaip(ki)
1Nminuskiminus1ρj(t) +
sumj aj [1minus p(kj)] Aij(t)kj
ρj(t) +sumjiajp(kj)
1Nminuskjminus1ρj(t)
Here j i indicates the sum over the nodes j not yet connected to i N minus kj(t)minus 1
is their number The quantity Aij(t) is the adjacency matrix of the time-integrated
network at time t ie it is equal to 1 if vi and vj have been in contact at least
once in the past and 0 otherwise In Eq (433) the only approximation made
is that the dynamical state of every node is considered to be independent of the
state of the partner in the interaction in other words we neglect the existence of
dynamical correlations among nodes which are created by the partially quenched
nature of the interaction pattern due to memory This is the same approximation
that is involved by the individual-based mean-eld approach for static networks
[93] discussed in Chapter 3
On the right hand side of Eq (433) the rst term is the recovery rate of ρi(t)
The second term describing the infection process is the product of λ times the
probability for vi to be susceptible and in curly brackets the fraction of infected
nodes contacted by vi per unit time In the curly brackets the rst and the second
term describe the case where vi is active and connects to the infected node vj taking
into account that the link can be an old or a new one respectively In the same
way the third and the fourth term represent the probabilities that vi is contacted
by an infected and active node vj
Since both Aij(t) and ki(t) depend on the evolution time t the behavior of the
epidemics can strongly depend on the starting time of the outbreak giving rise to
aging eects that will be investigated in numerical simulations When the epidemic
starts at very large times an analytic approach can be considered In this regime
with 1 ki(t) N we expect that the creation of new contacts can be ignored
and that the dynamical correlations are asymptotically negligible since the con-
nectivity of the integrated network becomes large If the epidemic starts at very
large times therefore we can apply an heterogeneous mean-eld approximation for
Aij(t) allowing for an analytical solution of the problem which we expect to be
asymptotically exact
The behavior for large times Lets consider the regime of large times where
1 ki(t) N for all nodes In this case each node has already had a large number
of contacts but that number is not too large so that the integrated network cannot
be considered as a complete graph ie it is still sparse In the limit of large N
there is a large temporal interval such that this condition is fullled The condition
1 ki(t) N allows us to replace in Eq (433) N minus ki(t)minus 1 with N and p(ki)
43 ANALYTICAL RESULTS 69
with (ki(t))minusβ Considering only leading terms Eq (433) becomes
(434) parttρi(t) = minusmicroρi(t) + λ [1minus ρi(t)]sumj
Aij(t)
(aiki
+ajkj
)ρj(t)
The linking probability To proceed further we perform the equivalent of the
heterogeneous mean-eld approximation for static networks ie we replace the
time-integrated adjacency matrix Aij(t) with its annealed form Pij(t) ie the
probability that vi and vj have been in contact in the past The evolution of Pij(t)
is described by the master equation
(435) parttPij(t) =
[aip(ki)
N minus ki minus 1+
ajpj(kj)
N minus kj minus 1
][1minus Pij(t)]
In Eq (435) Pij grows either because the node vi activates (probability per unit
time ai) it creates a new connection [probability p(ki)] and the new partner is
vi [probability (N minus ki minus 1)minus1] or because of the event with the role of vi and vjinterchanged
In the temporal interval of interest we can use again the relations holding for large
times p(ki) asymp kminusβi and N minus kj minus 1 asymp N Moreover for large times the degree of
a node of activity ai can be estimated by its average value k(ai t) given by Eq
(422) So we obtain
(436) parttPij(t) = [1minus Pij(t)]g(ai) + g(aj)
Ntβ
1+β
where we have dened
g(ai) = ai[C(ai)]β (437)
Eq (436) can be readily solved yielding
(438) Pij(t) = 1minus eminus(1+β)t1(1+β)
N [g(ai)+g(aj)]
In the regime t1(1+β) N Pij(t) becomes
(439) Pij(t) = (1 + β)t1(1+β)
N[g(ai) + g(aj)]
Notice that Pij(t) is a topological feature of the activity-driven network indepen-
dent of the epidemic process
Asymptotic ABMF equation We now introduce into Eq(434) the annealed
expression for the integrated adjacency matrix Aij(t) asymp Pij(t) = P (ai aj t) and
for the connectivity ki(t) = k(ai t) In this way the equations depend on the nodes
vi and vj only through their activities ai and aj The equation for the probability
ρ(a t) that a generic node of activity a is infected at time t is therefore
(4310)
parttρ(a t) = minusmicroρ(a t)+
λ [1minus ρ(a t)]
ag(a)g(a)+〈g〉
intdaprimeF (aprime)ρ(aprime t)+ a
g(a)+〈g〉intdaprimeF (aprime)ρ(aprime t)g(aprime)+
g(a)intdaprimeF (aprime) aprime
(g(aprime)+〈g〉)ρ(aprime t) +intdaprimeF (aprime) aprimeg(aprime)
(g(aprime)+〈g〉)ρ(aprime t)
43 ANALYTICAL RESULTS 70
where we have replaced the sums over nodes with integrals over the activities
1Nsumj rarr
intdaprimeF (aprime) and used Eq (423) which can be rewritten as
(4311) C(a) = (1 + β) [g(a) + 〈g〉]
Eq(4310) is eectively an ABMF approach since all the information on the be-
havior of the node vi depends on its activity ai Note that although Eqs (433)
and (434) described the dynamics of the individual node the further approxi-
mation underlying Eq (435) has transformed the approach into an ABMF one
conceptually analogous to the heterogeneous mean-eld approximation on static
networks where all the information on node vi is encoded in its degree ki
It is important to remark that in Eq (434) the time dependencies of P (ai aj t) propt1(1+β) and of the average degree k(ai t) prop t1(1+β) cancel out so that the right
hand side of Eq (4310) does not depend explicitly on time This suggests that in
this temporal regime the epidemic can be seen as an activity-driven process taking
place on an eective static graph where the probability for nodes vi and vj to be
linked is given by Eq (439) and the quantity t1(1+β)N is a xed quantity τ whose
value only determines the average degree of the network Performing simulations
over an ensemble of these eective static networks and averaging the results one
should then reproduce the predictions of the ABMF approach Eq (4310)
From Equation (4310) by performing a linear stability analysis around the absorb-
ing state ρ(a t) = 0 it is possible to compute analytically the epidemic threshold
(λmicro)c for any value of the reinforcement parameter β and of the exponent of the
analytical distribution ν Since for large times the node degrees diverge and cor-
relations can be neglected we expect the linear stability analysis to provide the
correct estimate of the epidemic threshold when the epidemics start at very long
times ie when the degrees ki(t) have already become very large
The results of the linear stability analysis are presented in Fig 431 showing that
the thresholds are smaller than in the memoryless case This lower value is a
consequence of the fact that memory reinforces the activity uctuations and in
these models uctuations clearly reduce the the epidemic threshold as shown by
Eq (432) The eect can be simply understood since nodes with large activity
have also a large degree therefore they are easily involved in epidemic contacts
not only because they are frequently activated but also because they are frequently
contacted by other nodes In this way memory reinforces the eect of activity
uctuations In this framework Fig 431 also shows that at large ν ie for in-
creasingly smaller uctuations the dierence with the memoryless model vanishes
In particular for F (a) = δ(a minus a0) ie when the activity does not uctuate one
obtains from Eq(4310) parttρ(t) = minusmicroρ(t) + 2a0λ[1minusρ(t)] that is the same equation
of the memoryless case This also explains the quite surprising observation that the
threshold is a growing function of β converging to the memoryless case as β rarrinfin
43 ANALYTICAL RESULTS 71
Figure 431 Plot of the ratio λcλML between the epidemicthresholds in the memory and in the memoryless (ML) casesfor dierent values of the exponent ν of the distribution F (a) =Baminus(ν+1) The dashed lines are the mean-eld memoryless resultswhile the solid lines are the outcomesnof the ABMF equations inpresence of memory
Indeed the tail of the degree distribution decays at large k as kminus[(1+β)ν+1] there-
fore at large β we get a faster decay and smaller degree uctuations For the same
reason in the limit β rarr 0 the dierence with the memoryless case is maximal since
degree inhomogeneities are stronger in this case
We remark that in Eq (4310) as in the memoryless case dynamical correlations
are ignored However we expect that at nite times due to the nite connectivity
of the integrated graph the eect of correlations becomes important The memory
process leads to the formation of small clusters of mutually connected high activity
vertices which become reservoirs of the disease in the SIS model The high fre-
quency of mutual contacts allows for reinfection favoring the overall survival of the
epidemic spreading in the system In this way social circles with high activity play
a role analogous to that played by the max K-core or the hub and its immediate
neighbors for SIS epidemics in static networks [26 27] To clarify the eect of dy-
namical correlations at nite time in the next Section we compare the analytical
predictions with results of numerical simulations As a nal remark we note that
in the asymptotic ABMF approach the linear stability analysis also holds for the
SIR model implying that the epidemic threshold is the same of the SIS model
However in the SIR model reinfection is not allowed so that the initial presence of
small clusters of mutually connected high activity vertices eectively inhibits the
spread of the disease For this reason we expect that nite connectivity (ie nite
time) increases the epidemic threshold with respect to the mean-eld result as we
will check in numerical simulations
45 NUMERICAL SIMULATIONS 72
44 Linear Stability Analysis
The dynamical process is described by the ABMF equation [Eq (4310)] which we
rewrite as
parttρ(a)(441)
λ [1minus ρ(a)] [A(a)g(a) 〈ρ(a)〉+A(a) 〈g(a)ρ(a)〉+ g(a) 〈A(a)ρ(a)〉+ 〈A(a)g(a)ρ(a)〉]
where for simplicity we have omitted the time dependencies and dened A(a) =
a[g(a) + 〈g(a)〉]
To study the stability of the system linearized around the xed point ρ(a) = 0 we
introduce the following functions
ρ = 〈ρ(a)〉x = 〈g(a)ρ(a)〉y = 〈A(a)ρ(a)〉z = 〈A(a)g(a)ρ(a)〉
Integrating Eq (441) over a and keeping only linear terms in ρ(a) we obtain an
equation for parttρ Similarly multiplying Eq (441) by g(a) and integrating over a
we get and equation for parttx Doing the same for y and z we obtain a closed system
of four equations for four variables
parttρ = minusmicroρ+ λ [〈A(a)g(a)〉 ρ+ 〈A(a)〉x+ 〈g(a)〉 y + z]
parttx = minusmicrox+ λ[langA(a)g2(a)
rangρ+ 〈A(a)g(a)〉x+
langg2(a)
rangy + 〈g(a)〉 z
]partty = minusmicroy + λ
[langA2(a)g(a)
rangρ+
langA2(a)
rangx+ 〈A(a)g(a)〉 y + 〈A(a)〉 z
]parttz = minusmicroz + λ
[langA2(a)g2(a)
rangρ+
langA2(a)g(a)
rangx+
langA(a)g2(a)
rangy + 〈A(a)g(a)〉 z
]These equations describe the epidemic near the state ρ(a) = 0 and the jacobian
matrix of this system of equations is
J =
λ 〈Ag〉 minus micro λ 〈A〉 λ 〈g〉 λ
λlangAg2
rangλ 〈Ag〉 minus micro λ
langg2rang
λ 〈g〉λlangA2g
rangλlangA2rang
λ 〈Ag〉 minus micro λ 〈A〉λlangA2g2
rangλlangA2g
rangλlangAg2
rangλ 〈Ag〉 minus micro
The state ρ(a) = 0 is stable provided all eigenvalues of J are negative hence the
epidemic threshold is given by the value (λmicro)c such that largest eigenvalue of the
Jacobian matrix is zero Numerical evaluation of the matrix J and of its eigenvalues
can be obtained rst by solving numerically Eq (423) to get C(a) and g(a) and
then calculating the averages dening J
45 Numerical simulations
SIS model on the eective static network As discussed above Eq (4310)
can be interpreted as a heterogeneous mean-eld equation for a SIS epidemic on an
45 NUMERICAL SIMULATIONS 73
10minus3 10minus2 10minus1⟨k⟩N
06
07
08
09
10
λcλ
ML
MLSimulations
Figure 451 Ratio between the epidemic threshold found in sim-ulations and the estimate given by equation Eq(432) valid for thememoryless model as a function of log(〈k〉N) For 〈k〉N gt 001we observe practically no dependence on 〈k〉
eective static network where the probability that vi and vj are connected is
(451) Pij = P (ai aj) = τ(1 + β)[g(ai) + g(aj)]
Here τ 1 is a constant g(a) = a[C(a)]β and C(a) is a function that can
be evaluated numerically for β gt 0 while for β = 0 it takes the simple form
C(a) = a+ 〈a〉 The constant τ can be tuned in order to set the average degree of
the network because
(452) k(a) = N
intdaprimeF (aprime)P (a aprime) = (1 + β)Nτ [g(a) + 〈g〉]
so that
(453) 〈k〉 =
intdaprimeF (aprime)k(aprime) = 2(1 + β)Nτ 〈g〉
We now study the SIS epidemic evolution on the eective static network
Given the activity of each node extracted according to the distribution F (a) for
each of the possible pairs of nodes we place an edge with probability given by
Eq (451) On top of this quenched topology we run a memoryless activity-driven
SIS dynamics starting with 10 of the nodes in the infected state until the sta-
tionary state is reached and we record the fraction of infected nodes We repeat
the procedure many times for each value of λ while micro is xed to 0015 We de-
termine the threshold as the position of the maximum of the susceptibility [39]
χ = N(ρ2 minus ρ2)ρ where the overbar denotes the average over dynamical real-
izations at xed topology We repeat all this for several networks obtained using
dierent sequences of activities and dierent samplings of Pij and we average the
epidemic thresholds found for each of them
45 NUMERICAL SIMULATIONS 74
Figure 452 Ratio between the epidemic threshold withmemory and the epidemic threshold of the memorylesscase as a function of the reinforcement parameter β =[001 02 04 06 1 14 18 22] for simulations on the eectivestatic network with ν = 24 ε = 001 N = 5 middot 104 The pointsare averages of dierent realizations of the network with dierentsequences of activity a1 a2 aN 32 realizations for 〈k〉 = 6 16realizations for 〈k〉 = 20 4 realizations for 〈k〉 = 100
We rst check that as long as 1 〈k〉 N the results are independent of the
exact value of 〈k〉 as predicted by the theory Fig 451 shows for β = 1 that the
eective threshold initially grows with 〈k〉 but then reaches a plateau in accordancewith the expectations
In Fig 452 we report the dependence of the eective epidemic threshold as a func-
tion of β for three values of the average degree 〈k〉 compared with the predictions
of the ABMF theory with and without memory We observe that as 〈k〉 growsnumerical results tend to coincide with theoretical predictions
On the other hand for small values of 〈k〉 the value of the threshold is smaller than
the one predicted theoretically Indeed on eective static networks with small
connectivity we expect the presence of clusters of mutually interconnected nodes
to be relevant as they are able to reinfect each other several times It is well
known that for the SIS model these backtracking eects tend to lower the epidemic
threshold since social circles with high activity favor the overall survival of the
epidemic
Epidemics on time-evolving networks Let us now consider simulations of
the epidemic spreading on the full time evolving network We consider a graph of
size N = 5 middot 104 with activity distributed according to F (a) = Baminus(ν+1) (ν = 24)
and a cuto ε = 10minus2 To extract the activities of individual nodes we perform an
importance sampling so that even in the nite size system the moments 〈a〉 andlanga2rangcoincide with their expected values
45 NUMERICAL SIMULATIONS 75
We rst start the temporal evolution of the network and at a later time t0 we let
the epidemic begin We evaluate at t0 the average connectivity of the nodes 〈k〉0which measures the evolution of the network at the starting time In both the SIS
and SIR models we use two dierent initial conditions The rst is to randomly
select (RA) the node to infect at time t0 Fig 454 and Fig 455 while the second
is to infect the most active node (MA) at time t0 We keep the recovery rate xed
at micro = 15 middot 10minus2 and vary the probability of infection λ to study the dependence
of its critical value on the memory parameter β
SIS model In the SIS model we determine the epidemic threshold using the
lifespan method We plot (see Fig 453) as a function of the parameter λ the
average lifespan of simulations ending before the coverage (ie the fraction of
distinct sites ever infected) reaches a preset value that we take equal to 12 The
threshold is estimated as the value of λ for which the lifespan has a peak
The epidemic thresholds of numerical simulations are compared with theoretical
predictions in Fig 454 (RA case) and 455 (MA case) Numerical results converge
toward the analytical prediction as 〈k〉0 becomes larger while there are strong
deviations for small 〈k〉0
For small 〈k〉0 two competing eects are at work First infections are mediated by
an eective static network with small connectivity therefore we expect backtracking
eects to enhance epidemic spreading and to lower the threshold However in this
case moves connecting new partners are also possible In these moves nodes are
chosen randomly in the whole system and the epidemic dynamics is memoryless
leading to a higher epidemic threshold So there exists a competition between
backtracking correlations and memoryless moves which reduce and increase the
threshold respectively Clearly for large 〈k〉0 both eects become negligible and
the ABMF result is recovered However at small β the memoryless moves are more
probable and indeed the threshold are larger while for large β memory eects are
more relevant We remark that the case β = 0 coincides with the memoryless case
(ML) and the dynamics never occurs on the eective static network On the other
hand for any β gt 0 at suciently large value of 〈k〉0 the dynamics is dominated
by memory and infections spread on the eective static network This originates a
singular behavior where the limits β rarr 0 and 〈k〉0 rarrinfin do not commute
Finally Figs 454 and 455 show that backtracking eects (leading to small thresh-
olds) are strong when the evolution starts from the most active site while they are
negligible with random initial conditions The most active node indeed has the
largest degree and it forms a cluster of highly activated nodes where the high
frequency of mutual contacts allows for reinfections and positive correlations Con-
versely the average site has a small connectivity and can activate new links with
high probability giving rise essentially to a memoryless epidemic dynamics
SIR model The results of simulations of the SIR process are displayed in Fig
456 and Fig 457 for the RA and MA case respectively The threshold is estimated
45 NUMERICAL SIMULATIONS 76
λλMF
L
⟨k⟩0=3⟨k⟩0=6⟨k⟩0=20⟨k⟩0=70⟨k⟩0=120
Figure 453 SIS epidemic process Lifespan (L) as function ofthe ratio between the epidemic threshold with memory and theepidemic threshold of the memoryless for dierent values of 〈k〉0N = 5 middot 104 ν = 24 a isin [10minus2 1] We consider 4 middot 103 epidemicrealizations for each value of λ Dynamics starts from the mostactive site and at small 〈k〉0 back-tracking eects are dominantfavoring the epidemic spreading this on one side lowers the valueof the threshold (value of λ corresponding to the peak) but alsoincreases the lifespan of the system at small λ
Figure 454 SIS epidemic process RA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
45 NUMERICAL SIMULATIONS 77
Figure 455 SIS epidemic process MA Ratio between the epi-demic threshold with memory and the epidemic threshold of thememoryless case as a function of the reinforcement parameterβ = [001 02 04 06 1 14 18 22] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) (WM) We consider 4 middot 103 epidemic realizations foreach value of λ
from the peak of the variability ∆ =radic〈N2
R〉 minus 〈NR〉2 〈NR〉 ie the standard
deviation of the number of recovered nodes NR at the end of the simulation As for
SIS at large 〈k〉0 dynamical correlations can be neglected and simulations recover
the ABMF result Simulations clearly show that now correlations at small 〈k〉0inhibit the epidemic spreading and the critical threshold becomes larger As in the
SIS model at small β the memory is negligible and the dynamics is driven by the
creation of new links so that the threshold values are close to the memoryless case
(ML) almost independently of 〈k〉0 On the other hand for larger β the epidemics
evolves on the integrated network dynamical correlations become important and
the thresholds grow even larger than in the memoryless case
45 NUMERICAL SIMULATIONS 78
Figure 456 SIR epidemic process RA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 2middot103 epidemicrealizations for each value of λ
Figure 457 SIR epidemic process MA Ratio between theepidemic threshold with memory and the epidemic threshold ofthe memoryless case as a function of the reinforcement param-eter β = [001 02 04 06 1 14 18] N = 5 middot 104 ν = 24a isin [10minus2 1] The dotted line represents the memoryless result(ML) the solid line is the analytical prediction obtained fromEq (4310) in the memory case (WM) We consider 4middot103 epidemicrealizations for each value of λ
CHAPTER 5
Conclusions
In this work we have stressed out the importance of time-varying networks when
processes evolve on the same time-scale of the evolving network in particular we
consider the activity-driven model In this framework we have considered a mech-
anism that emerge from the analysis of large dataset that can be interpreted as
the memory that individuals have of their pears in their social circle We have
presented our analytical and numerical results providing a complete understanding
of the interplay between the temporal evolution of the activity-driven network with
memory and the epidemic process occurring on top of it focusing our attention on
the SIS and SIR epidemic model To this end we have rst used an individual-based
mean-eld approach for the epidemic process on the activity-driven network with
memory keeping explicitly track of social contacts We have then studied the be-
havior for large time of the system hence we have considered the limit in which each
node has already had a large number of contacts but the time-integrated graph
can still be considered sparse In this regime we have performed an approximation
equivalent to the heterogeneous mean-eld for static networks and assigned the
description of the topology of the TVN to the linking probability that two nodes
have been in contact in the past At this point we have introduced the asymptotic
activity-based mean-eld description of the dynamics where all the information on
the behavior of the node depends on its activity In this framework the explicit time
dependency of the dynamical equation disappear and in this temporal regime the
epidemic can be considered as an activity-driven process on the eective static net-
work built with the linking probability Moreover in the case of large connectivity
correlations can be neglected and performing the linear stability analysis we have
obtained the epidemic threshold as a function of the memory parameter expected
to be exact at very long times
Comparing the analytic results with the numerical simulations on both the static
and the time-varying networks we have seen that the starting time of the outbreak
has crucial consequences on the epidemic threshold
In the long time limit the reinforcement mechanism of the topological evolution
completely inhibits the formation of new connections When the activity-driven
epidemic dynamics starts at this stage it takes place on a topology which can be
considered as static All nodes have a very large number of connections and hence
the mean-eld theory is asymptotically exact The epidemic threshold which is the
79
5 CONCLUSIONS 80
same for SIS and SIR dynamics is lower then the memoryless case because memory
enhances the eect of activity uctuations as also conrmed by the simulations
If instead the epidemic process starts before the memory has completely taken over
interesting model-dependent preasymptotic eects are observed The fundamental
observation is that at this stage nodes with large activity tend to interact with their
social circles while less active nodes still tend to explore the system creating new
connections The rst type of interaction tends to facilitate the spreading in the
SIS model while the second tends to suppress it This leads to positive or nega-
tive corrections to the asymptotic value of the threshold depending on the initial
conditions and on the reinforcement parameter β In the SIR case instead since
reinfection is not possible the interaction within social circles is strongly detrimen-
tal for the epidemic propagation so that the asymptotic value of the threshold is
always larger Hence our results allow to fully understand the contrasting eects
of strong ties on SIS and SIR dynamics observed in Ref [117]
Finally in this work we have fully explained the mechanism that the memory in-
troduces in the activity-driven network We have introduced an agile framework
to study spreading processes on highly correlated temporal network opening new
perspectives to control epidemic and information dynamics
Several possible extensions of the model considered here are possible to make it
more realistic both in terms of the topological evolution and of the spreading pro-
cess among them probably the most interesting would be the inclusion of burstiness
in agents activity We have seen the eect on activity-driven network of tie rein-
forcement and non exponentially-distributed inter-event times in Chapter 2 in this
framework the inclusion of a spreading dynamics is a promising and challenging
path for future research
Bibliography
[1] Lada A Adamic and Bernardo A Huberman The Webs Hidden Order
In Commun ACM 449 (Sept 2001) pp 5560 issn 0001-0782 doi 10
1145383694383707 url httpdoiacmorg101145383694
383707
[2] E Agliari et al Eciency of information spreading in a population of dif-
fusing agents In Phys Rev E 73 (4 Apr 2006) p 046138 doi 10
1103PhysRevE73046138 url httplinkapsorgdoi101103
PhysRevE73046138
[3] William Aiello Fan Chung and Linyuan Lu A random graph model for
power law graphs In Experiment Math 101 (2001) pp 5366 url http
projecteuclidorgeuclidem999188420
[4] Reacuteka Albert and Albert-Laacuteszloacute Barabaacutesi Statistical mechanics of complex
networks In Reviews of Modern Physics 741 (2002) pp 4797 issn 0034-
6861 doi 101103RevModPhys7447 arXiv 0106096v1 [arXivcond-mat]
url http dx doi org 10 1103 RevModPhys 74 47 7B 5C
7D5Cnpapers2publicationdoi101103RevModPhys74477B
5C7D5Cnhttpstacksioporg1478- 39751i=3a=006key=
crossref7e041937ef77358d60afe44f40925c5b7B5C7D5Cnhttp
linkapsorgdoi101103RevModPhys7447
[5] Reka Albert Hawoong Jeong and Albert-Laszlo Barabasi Internet Diam-
eter of the World-Wide Web In Nature 4016749 (Sept 1999) pp 130131
url httpdxdoiorg10103843601
[6] RM Anderson and RM May Infectious Diseases of Humans Dynamics
and Control Dynamics and Control OUP Oxford 1992 isbn 9780198540403
url httpsbooksgoogleitbooksid=HT0--xXBguQC
[7] Roland H Ggkvist Armen S Asratian Bipartite Graphs and Their Applica-
tions CAMBRIDGE UNIV PR May 11 2004 272 pp isbn 052159345X
url httpswwwebookdedeproduct4033924armen_s_asratian_
roland_h_ggkvist_bipartite_graphs_and_their_applicationshtml
[8] Norman T J Bailey Mathematical Theory of Epidemics Arnold 1957
isbn 0852641133 url httpwwwamazoncomMathematical-Theory-
Epidemics-Norman-Baileydp08526411333FSubscriptionId3D0JYN1NVW651KCA56C102
26tag3Dtechkie-2026linkCode3Dxm226camp3D202526creative
3D16595326creativeASIN3D0852641133
81
BIBLIOGRAPHY 82
[9] Paolo Bajardi et al Dynamical Patterns of Cattle Trade Movements In
PLoS ONE 65 (May 2011) e19869 doi 101371journalpone0019869
url httpdxdoiorg1013712Fjournalpone0019869
[10] D Balcan et al Multiscale mobility networks and the spatial spreading of
infectious diseases In PNAS 106 (2009) p 21484 doi 101073pnas
0906910106
[11] Albert Laacuteszloacute Barabaacutesi The origin of bursts and heavy tails in human
dynamics In Nature 4357039 (May 2005) pp 207211 issn 00280836
doi 101038nature03459 arXiv 0505371 [cond-mat] url http
wwwnaturecomarticlesnature03459
[12] A Barrat M Bartheacutelemy and A Vespignani Dynamical processes on com-
plex networks Cambridge University Press 2008 isbn 9780511791383 doi
10 1017 CBO9780511791383 url http dx doi org 10 1017
CBO9780511791383
[13] A Barrat et al The architecture of complex weighted networks In Pro-
ceedings of the National Academy of Sciences of the United States of Amer-
ica 10111 (Mar 2004) pp 374752 issn 0027-8424 doi 101073pnas
0400087101 url httpwwwncbinlmnihgovpubmed15007165
20httpwwwpubmedcentralnihgovarticlerenderfcgiartid=
PMC374315
[14] Marc Bartheacuteleacutemy and Lus A Nunes Amaral Small-World Networks Ev-
idence for a Crossover Picture In Phys Rev Lett 82 (15 Apr 1999)
pp 31803183 doi 101103PhysRevLett823180 url httplink
apsorgdoi101103PhysRevLett823180
[15] Marc Bartheacutelemy et al Dynamical patterns of epidemic outbreaks in com-
plex heterogeneous networks In Journal of Theoretical Biology 2352 (2005)
pp 275288 issn 00225193 doi 101016jjtbi200501011 arXiv
0410330 [cond-mat]
[16] Marc Bartheacutelemy et al Velocity and Hierarchical Spread of Epidemic Out-
breaks in Scale-Free Networks In Physical Review Letters 9217 (Apr
2004) doi 101103physrevlett92178701
[17] S Blanes et al The Magnus expansion and some of its applications In
Physics Reports 4705-6 (Jan 2009) pp 151238 doi 101016jphysrep
200811001
[18] Stefano Boccaletti et al Complex networks Structure and dynamics 2006
doi 101016jphysrep200510009 arXiv 161001345 url www
elseviercomlocatephysrep
[19] Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Epidemic spreading in cor-
related complex networks In Physical Review E - Statistical Physics Plas-
mas Fluids and Related Interdisciplinary Topics 664 (2002) p 4 issn
1063651X doi 101103PhysRevE66047104 arXiv 0205621 [cond-mat]
BIBLIOGRAPHY 83
url httpsjournalsapsorgprepdf101103PhysRevE66
047104
[20] B Bollobas Random Graphs Ed by W Fulton et al Cambridge University
Press 2001
[21] Bela Bollobas Modern Graph Theory Springer 1998
[22] Andrei Broder et al Graph Structure in the Web In Comput Netw 331-6
(June 2000) pp 309320 issn 1389-1286 doi 101016S1389-1286(00)
00083-9 url httpdxdoiorg101016S1389-1286(00)00083-9
[23] Raaella Burioni Enrico Ubaldi and Alessandro Vezzani Asymptotic the-
ory of time varying networks with burstiness and heterogeneous activation
patterns In J Stat Mech Theory Exp 20175 (May 2017) p 054001 issn
17425468 doi 1010881742-5468aa6ce7 url httpstacksiop
org1742-54682017i=5a=054001key=crossrefe5f692cb999d11ff344d124f66866b45
[24] G Caldarelli et al Scale-Free Networks from Varying Vertex Intrinsic Fit-
ness In Phys Rev Lett 89 (25 Dec 2002) p 258702 doi 101103
PhysRevLett89258702 url httplinkapsorgdoi101103
PhysRevLett89258702
[25] Guido Caldarelli Scale-Free Networks 2007 doi 101093acprofoso
97801992115170010001
[26] Claudio Castellano and Romualdo Pastor-Satorras Competing activation
mechanisms in epidemics on networks In Sci Rep 2 (Apr 2012) p 371
doi 101038srep00371
[27] Claudio Castellano and Romualdo Pastor-Satorras Relating Topological
Determinants of Complex Networks to Their Spectral Properties Structural
and Dynamical Eects In Phys Rev X 7 (4 Oct 2017) p 041024 doi
101103PhysRevX7041024 url httpslinkapsorgdoi10
1103PhysRevX7041024
[28] Claudio Castellano and Romualdo Pastor-Satorras Thresholds for epidemic
spreading in networks In Physical Review Letters 10521 (2010) pp 14
issn 00319007 doi 101103PhysRevLett105218701 arXiv 1010
1646
[29] Ciro Cattuto et al Dynamics of Person-to-Person Interactions from Dis-
tributed RFID Sensor Networks In PLoS ONE 57 (July 2010) e11596
doi 101371journalpone0011596 url httpdxdoiorg10
13712Fjournalpone0011596
[30] F Chung L Lu and V Vu Spectra of random graphs with given expected
degrees In Proceedings of the National Academy of Sciences 10011 (May
2003) pp 63136318 doi 101073pnas0937490100
[31] Aaron Clauset Cosma Rohilla Shalizi and M E J Newman Power-Law
Distributions in Empirical Data In SIAM Review 514 (2009) pp 661703
issn 0036-1445 doi 101137070710111 arXiv arXiv07061062v2
url https arxiv org pdf 0706 1062 pdf 20http epubs
BIBLIOGRAPHY 84
siamorgdoiabs1011370707101117B5C7D5Cnpapers3
publicationdoi10113707071011120httpsepubssiamorg
doipdf10113707071011120httpepubssiamorgdoi101137
07071011120httpepubssiamorgdoipdf1
[32] DR Cox Renewal Theory Methuen science paperbacks Methuen 1970
isbn 9780416523805 url httpsbooksgoogleitbooksid=cClHYAAACAAJ
[33] Easley David and Kleinberg Jon Networks Crowds and Markets Reasoning
About a Highly Connected World New York NY USA Cambridge Univer-
sity Press 2010 isbn 0521195330 9780521195331
[34] O Diekmann J Heersterbeek and J Metz On the denition and the
computation of the basic reproduction number ratio R0 in models for infec-
tious diseases in heterogeneous populations In J Math Biol 28 (1990) doi
101007BF00178324 url httpdxdoiorg101007BF00178324
[35] S N Dorogovtsev A V Goltsev and J F F Mendes Critical phenomena
in complex networks In Rev Mod Phys 80 (4 Oct 2008) pp 12751335
doi 101103RevModPhys801275 url httpslinkapsorgdoi
101103RevModPhys801275
[36] R I M Dunbar Neocortex size as a constraint on group size in primates
In J Hum Evol 226 (1992) pp 469493 doi 1010160047-2484(92)
90081-J url httpwwwsciencedirectcomsciencearticlepii
004724849290081J
[37] Paul Erdos and Alfred Renyi On the evolution of random graphs In Publ
Math Inst Hungary Acad Sci 5 (1960) pp 1761
[38] S Eubank et al Modelling disease outbreaks in realistic urban social net-
works In Nature 429 (2004) p 180 doi 101038nature02541 url
httpdxdoiorg101038nature02541
[39] Silvio C Ferreira Claudio Castellano and Romualdo Pastor-Satorras Epi-
demic thresholds of the susceptible-infected-susceptible model on networks
A comparison of numerical and theoretical results In Phys Rev E 86 (4
Oct 2012) p 041125 doi 101103PhysRevE86041125 url https
linkapsorgdoi101103PhysRevE86041125
[40] Laetitia Gauvin et al Randomized reference models for temporal networks
In (June 11 2018) arXiv httparxivorgabs180604032v1 [physicssoc-ph]
[41] S Gil and D H Zanette Optimal disorder for segregation in annealed small
worlds In The European Physical Journal B 472 (Sept 2005) pp 265
273 doi 101140epjbe2005-00319-8
[42] K-I Goh and A-L Barabaacutesi Burstiness and memory in complex systems
In EPL (Europhysics Letters) 814 (2008) p 48002 url httpstacks
ioporg0295-507581i=4a=48002
[43] S Goacutemez et al Discrete-time Markov chain approach to contact-based dis-
ease spreading in complex networks In EPL (Europhysics Letters) 893
(Feb 2010) p 38009 doi 1012090295-50758938009
BIBLIOGRAPHY 85
[44] Bruno Gonccedilalves Nicola Perra and Alessandro Vespignani Modeling Usersampapos
Activity on Twitter Networks Validation of Dunbarampaposs Number In
Plos One 68 (2011) e22656 issn 1932-6203 doi 101371journalpone
0022656 url httpdxplosorg101371journalpone0022656
[45] Mark S Granovetter The Strength of Weak Ties In Am J Sociol 786
(1973) pp 13601380 issn 00029602 15375390 url httpwwwjstor
orgstable2776392
[46] M Henkel et al Non-Equilibrium Phase Transitions Volume 2 Ageing
and Dynamical Scaling Far from Equilibrium Theoretical and Mathemat-
ical Physics Springer 2011 isbn 9789048128693 url httpsbooks
googlefrbooksid=AiofeEteLVcC
[47] Malte Henkel and Michel Pleimling Non-Equilibrium Phase Transitions
Volume 2 Ageing and Dynamical Scaling Far from Equilibrium Springer
Netherlands 2010 isbn 9789048128693 doi 101007978-90-481-2869-
3
[48] Herbert W Hethcote The Mathematics of Infectious Diseases In SIAM
Review 424 (Jan 2000) pp 599653 issn 0036-1445 doi 10 1137
S0036144500371907 url httpwwwsiamorgjournalsojsaphp
20http leonidzhukov net hse 2014 socialnetworks papers
2000SiamRevpdf20httpepubssiamorgdoi101137S0036144500371907
[49] P Holme and J Saramaumlki Temporal networks In Phys Rep 5193 (2012)
pp 97125 issn 0370-1573 doi 101016jphysrep201203001 url
httpwwwsciencedirectcomsciencearticlepiiS0370157312000841
[50] Petter Holme and Jari Saramaumlki Temporal Networks In (2011) pp 128
issn 03701573 doi 101016jphysrep201203001 arXiv 11081780
url httparxivorgabs110817807B5C7D0Ahttpdxdoi
org101016jphysrep201203001
[51] Stanley (University of Illinois Urbana-Champaign) Wasserman and Kather-
ine (University of South Carolina) Faust Structural Analysis in the So-
cial Sciences Cambridge University Press Nov 25 1994 857 pp isbn
0521387078 url httpswwwebookdedeproduct3719721stanley_
university_of_illinois_urbana_champaign_wasserman_katherine_
university_of_south_carolina_faust_structural_analysis_in_the_
social_scienceshtml
[52] N G van Kampen Stochastic Processes in Physics and Chemistry Elsevier
LTD Oxford Mar 11 2007 464 pp isbn 0444529659 url https
wwwebookdedeproduct6071989n_g_van_kampen_stochastic_
processes_in_physics_and_chemistryhtml
[53] M Karsai et al Small but slow world How network topology and burstiness
slow down spreading In Phys Rev E 83 (2 Feb 2011) p 025102 doi 10
1103 url httpslinkapsorgdoi101103PhysRevE83025102
BIBLIOGRAPHY 86
[54] Maacuterton Karsai Kimmo Kaski and Jaacutenos Kerteacutesz Correlated dynamics in
egocentric communication networks In PLoS One 77 (July 2012) Ed by
Renaud Lambiotte e40612 issn 19326203 doi 101371journalpone
0040612 url httpdxplosorg101371journalpone0040612
[55] Maacuterton Karsai Nicola Perra and Alessandro Vespignani Time varying
networks and the weakness of strong ties In Sci Rep 4 (Feb 2014) p 4001
url httpdxdoiorg101038srep04001
[56] Matt J Keeling Modeling Infectious Diseases in Humans and Animals
Princeton University Press Oct 11 2007 isbn 0691116172 url https
wwwebookdedeproduct6599625matt_j_keeling_modeling_
infectious_diseases_in_humans_and_animalshtml
[57] Matt J Keeling and Ken TD D Eames Networks and epidemic models
In Journal of The Royal Society Interface 24 (Sept 2005) pp 295307
issn 1742-5689 doi 101098rsif20050051 url httprsif
royalsocietypublishingorgcgidoi101098rsif20050051
20httprsifroyalsocietypublishingorgcontentroyinterface
24295fullpdf
[58] W O Kermack and A G McKendrick A Contribution to the Mathemati-
cal Theory of Epidemics In Proceedings of the Royal Society of London A
Mathematical Physical and Engineering Sciences 115772 (1927) pp 700
721 issn 0950-1207 doi 101098rspa19270118 eprint httprspa
royalsocietypublishingorgcontent115772700fullpdf url
httprsparoyalsocietypublishingorgcontent115772700
[59] Hyewon Kim Meesoon Ha and Hawoong Jeong Dynamic topologies of
activity-driven temporal networks with memory In Phys Rev E 97 (6
June 2018) p 062148 doi 101103PhysRevE97062148 url https
linkapsorgdoi101103PhysRevE97062148
[60] Hyewon Kim Meesoon Ha and Hawoong Jeong Scaling properties in time-
varying networks with memory In Eur Phys J B 8812 (Dec 2015)
p 315 issn 1434-6036 doi 101140epjbe2015-60662-7 url https
doiorg101140epjbe2015-60662-7
[61] Gueorgi Kossinets Jon Kleinberg and Duncan Watts The structure of
information pathways in a social communication network In Proceeding of
the 14th ACM SIGKDD international conference on Knowledge discovery
and data mining - KDD 08 ACM Press 2008 doi 1011451401890
1401945
[62] Renaud Lambiotte Vsevolod Salnikov and Martin Rosvall Eect of mem-
ory on the dynamics of random walks on networks In J Complex Networks
32 (June 2015) pp 177188 issn 20511329 doi 101093comnetcnu017
url httpsacademicoupcomcomnetarticle- lookupdoi10
1093comnetcnu017
BIBLIOGRAPHY 87
[63] Vito Latora and Massimo Marchiori Ecient Behavior of Small-World
Networks In Physical Review Letters 8719 (Oct 2001) doi 101103
physrevlett87198701
[64] Suyu Liu et al Controlling Contagion Processes in Activity Driven Net-
works In Phys Rev Lett 112 (11 Mar 2014) p 118702 doi 101103
PhysRevLett112118702 url httplinkapsorgdoi101103
PhysRevLett112118702
[65] Roberto Livi and Paolo Politi Nonequilibrium Statistical Physics Cam-
bridge University Press Oct 2017 doi 1010179781107278974
[66] JOSEPH J LUCZKOVICH et al Dening and Measuring Trophic Role
Similarity in Food Webs Using Regular Equivalence In Journal of Theoret-
ical Biology 2203 (Feb 2003) pp 303321 doi 101006jtbi20033147
[67] Naoki Masuda and Renaud Lambiotte A Guide to Temporal Networks
WORLD SCIENTIFIC (EUROPE) Apr 2016 doi 101142q0033
[68] Naoki Masuda Mason A Porter and Renaud Lambiotte Random walks and
diusion on networks 2017 doi 101016jphysrep201707007 arXiv
161203281 url wwwelseviercomlocatephysrep
[69] P Van Mieghem and R van de Bovenkamp Non-Markovian Infection
Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic
Threshold in Networks In Physical Review Letters 11010 (Mar 2013)
doi 101103physrevlett110108701
[70] P Van Mieghem J Omic and R Kooij Virus Spread in Networks In
IEEEACM Transactions on Networking 171 (Feb 2009) pp 114 doi
101109tnet2008925623
[71] Joel C Miller Epidemic size and probability in populations with heteroge-
neous infectivity and susceptibility In Physical Review E 761 (July 2007)
doi 101103physreve76010101
[72] Giovanna Miritello et al Limited communication capacity unveils strategies
for human interaction In Sci Rep 3 (2013) p 1950 url httpdx
doiorg101038srep01950
[73] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Phys Rev Lett 114 (10
Mar 2015) p 108701 doi 10 1103 PhysRevLett 114 108701 url
httplinkapsorgdoi101103PhysRevLett114108701
[74] Antoine Moinet Michele Starnini and Romualdo Pastor-Satorras Bursti-
ness and Aging in Social Temporal Networks In Physical Review Letters
11410 (Mar 2015) p 108701 issn 10797114 doi 101103PhysRevLett
114108701 arXiv 14120587 url httpsjournalsapsorgprl
pdf101103PhysRevLett11410870120httpslinkapsorgdoi
101103PhysRevLett114108701
BIBLIOGRAPHY 88
[75] MICHAEL MOLLOY and BRUCE REED The Size of the Giant Com-
ponent of a Random Graph with a Given Degree Sequence In Combi-
natorics Probability and Computing 7 (03 Sept 1998) pp 295305 issn
1469-2163 doi null url httpjournalscambridgeorgarticle_
S0963548398003526
[76] Yamir Moreno Romualdo Pastor-Satorras and Alessandro Vespignani Epi-
demic outbreaks in complex heterogeneous networks In European Physical
Journal B 264 (2002) pp 521529 issn 14346028 doi 101007s10051-
002-8996-y arXiv 0107267 [cond-mat] url httpsarxivorgpdf
cond-mat0107267pdf
[77] Matthieu Nadini et al Epidemic spreading in modular time-varying net-
works In Scientic Reports 81 (Feb 2018) doi 101038s41598-018-
20908-x
[78] M E J Newman Spread of epidemic disease on networks In Phys Rev
E 66 (1 July 2002) p 016128 doi 101103PhysRevE66016128 url
httplinkapsorgdoi101103PhysRevE66016128
[79] M E J Newman The Structure and Function of Complex Networks In
SIAM Review 452 (2003) pp 167256 doi 101137S003614450342480
eprint httpdxdoiorg101137S003614450342480 url http
dxdoiorg101137S003614450342480
[80] M E J Newman The Structure and Function of Complex Networks
In SIAM Review 452 (Jan 2003) pp 167256 issn 0036-1445 doi 10
1137S003614450342480 url httpepubssiamorgdoipdf10
1137S00361445034248020httpepubssiamorgdoi101137
S003614450342480
[81] M E J Newman I Jensen and R M Zi Percolation and epidemics in a
two-dimensional small world In Phys Rev E 65 (2 Jan 2002) p 021904
doi 101103PhysRevE65021904 url httplinkapsorgdoi10
1103PhysRevE65021904
[82] M E J Newman S H Strogatz and D J Watts Random graphs with
arbitrary degree distributions and their applications In Phys Rev E 64
(2 July 2001) p 026118 doi 101103PhysRevE64026118 url http
linkapsorgdoi101103PhysRevE64026118
[83] Mark Newman Networks An Introduction In Networks An Introduction
(2010) pp 1784 issn 1578-1275 doi 101093acprofoso9780199206650
0010001 arXiv 12122425
[84] Vincenzo Nicosia et al Components in time-varying graphs In journal of
nonlinear (2012) arXiv arXiv11062134v3 url httpsarxiv
orgpdf11062134pdf20httpaipscitationorgdoiabs10
106313697996
BIBLIOGRAPHY 89
[85] Vincenzo Nicosia et al Temporal Networks In (2013) issn 1860-0832
doi 101007978-3-642-36461-7 url httplinkspringercom
101007978-3-642-36461-7
[86] Jae Dong Noh and Heiko Rieger Random Walks on Complex Networks
In Physical Review Letters 9211 (Mar 2004) doi 101103physrevlett
92118701
[87] Tore Opsahl et al Prominence and Control The Weighted Rich-Club Ef-
fect In Physical Review Letters 10116 (Oct 2008) doi 101103physrevlett
101168702
[88] Raj Kumar Pan and Jari Saramaumlki Path lengths correlations and cen-
trality in temporal networks In Phys Rev E 84 (1 July 2011) p 016105
doi 101103PhysRevE84016105 url httplinkapsorgdoi10
1103PhysRevE84016105
[89] R Pastor-Satorras and A Vespignani Epidemic spreading in scale-free net-
works In Phys Rev Lett 86 (2001) p 3200 doi 101103PhysRevLett
863200 url httpdxdoiorg101103PhysRevLett863200
[90] Romualdo Pastor-Satorras Alexei Vaacutezquez and Alessandro Vespignani Dy-
namical and Correlation Properties of the Internet In Phys Rev Lett 87
(25 Nov 2001) p 258701 doi 101103PhysRevLett87258701 url
httplinkapsorgdoi101103PhysRevLett87258701
[91] Romualdo Pastor-Satorras and Alessandro Vespignani Epidemic spread-
ing in scale-free networks In Physical Review Letters 8614 (Apr 2001)
pp 32003203 issn 00319007 doi 101103PhysRevLett863200 arXiv
0010317 [cond-mat] url httpsjournalsapsorgprlpdf10
1103PhysRevLett86320020httpslinkapsorgdoi101103
PhysRevLett863200
[92] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Reviews of Modern Physics 873 (2015) pp 925979 issn 15390756
doi 101103RevModPhys87925 arXiv 14082701
[93] Romualdo Pastor-Satorras et al Epidemic processes in complex networks
In Rev Mod Phys 87 (3 Aug 2015) pp 925979 doi 101103RevModPhys
87925 url httpslinkapsorgdoi101103RevModPhys87925
[94] N Perra et al Activity driven modeling of time varying networks In Sci
Rep 21 (Dec 2012) p 469 issn 2045-2322 doi 101038srep00469
url httpwwwnaturecomarticlessrep00469
[95] N Perra et al Random Walks and Search in Time-Varying Networks
In Phys Rev Lett 10923 238701 (Dec 2012) p 238701 doi 101103
PhysRevLett109238701
[96] Nicola Perra et al Random walks and search in time-varying networks
In Physical Review Letters 10923 (2012) pp 15 issn 00319007 doi
101103PhysRevLett109238701 arXiv 12062858 url https
journalsapsorgprlpdf101103PhysRevLett109238701
BIBLIOGRAPHY 90
[97] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In (2017) arXiv 1703
02482 url httparxivorgabs170302482
[98] Iacopo Pozzana Kaiyuan Sun and Nicola Perra Epidemic Spreading on
Activity-Driven Networks with Attractiveness In Phys Rev E 964 (2017)
p 042310 doi 101103PhysRevE96042310
[99] Alessandro Rizzo and Maurizio Porri Innovation diusion on time-varying
activity driven networks In Eur Phys J B 891 (Jan 2016) p 20 issn
14346036 doi 10 1140 epjb e2015 - 60933 - 3 url http link
springercom101140epjbe2015-60933-3
[100] Martin Rosvall et al Memory in network ows and its eects on spreading
dynamics and community detection In Nat Commun 51 (Dec 2014)
p 4630 issn 20411723 doi 101038ncomms5630 url httpwww
naturecomarticlesncomms5630
[101] Jari Saramaumlki and Esteban Moro From seconds to months an overview of
multi-scale dynamics of mobile telephone calls In Eur Phys Jour B 886
(2015) p 164 url httpdxdoiorg101140epjbe2015-60106-6
[102] Jari Saramaumlki et al Persistence of social signatures in human communica-
tion In PNAS 1113 (2014) pp 942947 issn 0027-8424 doi 101073
pnas1308540110
[103] Ingo Scholtes et al Causality-driven slow-down and speed-up of diusion in
non-Markovian temporal networks In Nat Commun 51 (2014) p 5024
issn 2041-1723 doi 101038ncomms6024 url httpwwwnature
com doifinder 10 1038 ncomms6024 20http www nature com
articlesncomms6024
[104] M Aacutengeles Serrano Mariaacuten Boguntildeaacute and Romualdo Pastor-Satorras Cor-
relations in weighted networks In Phys Rev E 74 (5 Nov 2006) p 055101
doi 101103PhysRevE74055101 url httplinkapsorgdoi10
1103PhysRevE74055101
[105] Kieran J Sharkey Deterministic epidemic models on contact networks
Correlations and unbiological terms In Theoretical Population Biology 794
(June 2011) pp 115129 doi 101016jtpb201101004
[106] Shigeru Shinomoto Keiji Miura and Shinsuke Koyama A measure of local
variation of inter-spike intervals In Biosystems 791-3 (Jan 2005) pp 67
72 doi 101016jbiosystems200409023
[107] American Physical Society APS Data Sets for Research url https
journalsapsorgdatasets
[108] Bo Soumlderberg General formalism for inhomogeneous random graphs In
Phys Rev E 66 (6 Dec 2002) p 066121 doi 101103PhysRevE66
066121 url httplinkapsorgdoi101103PhysRevE66066121
[109] D J de Solla Price Networks of Scientic Papers In Science 1493683
(July 1965) pp 510515 doi 101126science1493683510
BIBLIOGRAPHY 91
[110] Leo Speidel et al Temporal interactions facilitate endemicity in the susceptible-
infected-susceptible epidemic model In New Journal of Physics 187 (July
2016) pp 124 issn 13672630 doi 1010881367-2630187073013
arXiv 160200859 url httpstacksioporg1367-263018i=7a=
073013key=crossref950461fc36231b959b6c696b8c789b92
[111] Michele Starnini and Romualdo Pastor-Satorras Temporal percolation in
activity-driven networks In Phys Rev E 89 (3 Mar 2014) p 032807
doi 101103PhysRevE89032807 url httplinkapsorgdoi10
1103PhysRevE89032807
[112] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 876 (June 2013) p 062807 issn 1539-
3755 doi 101103PhysRevE87062807 arXiv arXiv13043370v1
url httpslinkapsorgdoi101103PhysRevE87062807
20httpsjournalsapsorgprepdf101103PhysRevE87062807
[113] Michele Starnini and Romualdo Pastor-Satorras Topological properties of a
time-integrated activity-driven network In Phys Rev E 87 (6 June 2013)
p 062807 doi 101103PhysRevE87062807 url httplinkaps
orgdoi101103PhysRevE87062807
[114] Dietrich Stauer and Muhammad Sahimi Diusion in scale-free networks
with annealed disorder In Physical Review E 724 (Oct 2005) doi 10
1103physreve72046128
[115] Juliette Stehleacute Alain Barrat and Ginestra Bianconi Dynamical and bursty
interactions in social networks In Phys Rev E 81 (3 Mar 2010) p 035101
doi 101103PhysRevE81035101 url httplinkapsorgdoi10
1103PhysRevE81035101
[116] James Stiller and R I M Dunbar Perspective-taking and memory capacity
predict social network size In Soc Netw 291 (Jan 2007) pp 93104 doi
101016jsocnet200604001 url httpwwwsciencedirectcom
sciencearticlepiiS0378873306000128
[117] Kaiyuan Sun Andrea Baronchelli and Nicola Perra Contrasting eects of
strong ties on SIR and SIS processes in temporal networks In Eur Phys J
B 8812 (2015) p 326 issn 14346036 doi 101140epjbe2015-60568-4
[118] Michele Tizzani et al Epidemic Spreading and Aging in Temporal Networks
with Memory In (July 31 2018) arXiv httparxivorgabs1807
11759v2 [physicssoc-ph]
[119] Jerey Travers and Stanley Milgram An Experimental Study of the Small
World Problem In Sociometry 324 (Dec 1969) p 425 doi 102307
2786545
[120] Enrico Ubaldi et al Asymptotic theory for the dynamic of networks with
heterogenous social capital allocation In Scientic Reports 6 (2015) p 35724
doi 101038srep35724 arXiv 150904563 url httpwwwnature
BIBLIOGRAPHY 92
comarticlessrep357247B5C7D5Cnhttparxivorgabs1509
04563
[121] Enrico Ubaldi et al Asymptotic theory of time-varying social networks
with heterogeneous activity and tie allocation In Sci Rep 6 (Oct 2016)
p 35724 url httpdxdoiorg101038srep35724
[122] Enrico Ubaldi et al Burstiness and tie activation strategies in time-varying
social networks In Scientic Reports 7 (2017) issn 20452322 doi 10
1038srep46225 arXiv 160708910 url httpswwwnaturecom
articlessrep46225pdf
[123] Eugenio Valdano et al Analytical computation of the epidemic threshold
on temporal networks In Physical Review X 52 (2015) pp 19 issn
21603308 doi 101103PhysRevX5021005 arXiv 14064815
[124] Eugenio Valdano et al Epidemic Threshold in Continuous-Time Evolving
Networks In Phys Rev Lett 120 (6 Feb 2018) p 068302 doi 101103
PhysRevLett120068302 url httpslinkapsorgdoi101103
PhysRevLett120068302
[125] Piet Van Mieghem and Eric Cator Epidemics in networks with nodal self-
infection and the epidemic threshold In Physical Review E - Statistical
Nonlinear and Soft Matter Physics 861 (2012) issn 15393755 doi 10
1103PhysRevE86016116 url httpsjournalsapsorgprepdf
101103PhysRevE86016116
[126] Piet Van Mieghemy Faryad Darabi Sahnehz and Caterina Scoglioz An
upper bound for the epidemic threshold in exact Markovian SIR and SIS
epidemics on networks In Proceedings of the IEEE Conference on Deci-
sion and Control Vol 2015-Febru February 2014 pp 62286233 isbn
9781467360883 doi 101109CDC20147040365 arXiv 14021731 url
httpsarxivorgpdf14021731pdf
[127] Alexei Vaacutezquez Romualdo Pastor-Satorras and Alessandro Vespignani Large-
scale topological and dynamical properties of the Internet In Phys Rev
E 65 (6 June 2002) p 066130 doi 101103PhysRevE65066130 url
httplinkapsorgdoi101103PhysRevE65066130
[128] Alexei Vaacutezquez et al Modeling bursts and heavy tails in human dynamics
In Phys Rev E 73 (3 Mar 2006) p 036127 doi 101103PhysRevE73
036127 url httplinkapsorgdoi101103PhysRevE73036127
[129] A Vespignani M Bathelemy and A BarratDynamical Processes on Complex
Networks 2008 p 367 isbn 9780521879507 doi 101017CBO9780511791383
arXiv arXiv10111669v3
[130] Thomas Andrew Waigh The Physics of Living Processes A Mesoscopic
Approach PAPERBACKSHOP UK IMPORT Oct 20 2014 599 pp isbn
1118449940 url httpswwwebookdedeproduct22523623thomas_
andrew_waigh_the_physics_of_living_processes_a_mesoscopic_
approachhtml
BIBLIOGRAPHY 93
[131] Yang Wang et al Epidemic spreading in real networks an eigenvalue view-
point In 22nd International Symposium on Reliable Distributed Systems
2003 Proceedings IEEE Comput Soc doi 101109reldis20031238052
[132] Duncan J Watts and Steven H Strogatz Collective dynamics of `small-
world networks In Nature 3936684 (June 1998) pp 440442 url http
dxdoiorg10103830918
[133] J M Yeomans Statistical Mechanics of Phase Transitions OXFORD UNIV
PR May 11 1992 168 pp isbn 0198517300 url httpswwwebook
dedeproduct3596629j_m_yeomans_statistical_mechanics_of_
phase_transitionshtml